Dynamic modelling and advanced control analysis are employed to investigate the behavior of a self-balancing electric Segway navigating on irregular sinusoidal terrains. A nonlinear dynamic model treats the Segway as a cart–inverted pendulum system, incorporating sinusoidal road irregularities and elastic and damping interactions at the wheel–ground interface. Equations of motion are derived using the Lagrangian formulation and linearized around equilibrium with Taylor expansions. Numerical simulations via the fourth-order Runge–Kutta method reveal significant increases in vibration amplitudes and system sensitivity at higher speeds. Kernel Density Estimation (KDE) is applied to translational and angular vibration data, yielding smooth distributions, while Lorenz-like attractors indicate deterministic chaos under certain excitations. Stability is assessed through bifurcation analysis, which reveals a critical forward velocity of 20.102 m/s, beyond which the system transitions into instability. The findings also identify critical thresholds, including a friction-to-mass ratio above approximately 7, a gravity-to-length ratio exceeding 25, and a mass ratio near 0.3. Each of them contributes to heightened instability and complex oscillatory responses. The significant influence of irregular terrain on system dynamics underscores the necessity for robust control strategies to maintain operational stability and rider comfort on uneven surfaces.
Ntambo et al. (2026) studied this question.