<p>Owing to the intricate bifurcation behaviors of inverted pendulum systems, accurate and efficient semi-analytical computation of periodic responses and their stability assessment remain important yet challenging. An efficient and robust incremental harmonic balance (ER-IHB) method is developed in this study to compute periodic and potential period-doubling responses and to conduct bifurcation analysis of inverted pendulum systems. The fast Fourier transform is employed to compute the residuals and Jacobian matrices efficiently, and multiple search directions are adaptively combined to improve convergence robustness. A path-following continuation scheme is implemented to trace the response curves versus both excitation frequency and amplitude. The Floquet theory is applied to assess the stability of solutions and perform bifurcation analysis. The effectiveness of the ER-IHB method is demonstrated by two inverted pendulum systems : (i) a system with a cubic stiffness nonlinearity and a vertically oscillating base, and (ii) a system with a horizontally moving base mass subjected to harmonic forcing. The results show excellent agreement with those obtained by the fourth-order Runge-Kutta method, and multiple bifurcations are accurately identified, including saddle-node, symmetry-breaking pitchfork, period-doubling, and Neimark-Sacker bifurcations.</p>