Energy-based analysis of self-excited limit cycles on nonlinear modal manifolds in a variable-stiffness tensegrity system

Self-excited oscillations provide an effective means of exploiting the intrinsic dynamics of tensegrity systems for motion generation. However, for variable-stiffness tensegrity systems undergoing large-amplitude motion, conventional linear modal analysis is insufficient to predict the existence and properties of limit-cycle oscillations. This paper develops an energy-based analytical framework for self-excited limit cycles on nonlinear modal manifolds in a variable-stiffness tensegrity system. A two-degree-of-freedom tensegrity robotic leg is adopted as the study platform. The governing dynamic equations are established, and periodic orbits of the corresponding conservative system are computed on nonlinear modal manifolds using a Poincaré map-based continuation method. By evaluating the energy variation along the family of periodic orbits of the conservative system, an energy-based criterion is established to determine the existence and stability of self-excited limit cycles. Numerical results show that the proposed framework successfully predicts stable limit cycles and captures the amplitude-dependent evolution of periodic orbits and oscillation frequencies in the nonlinear regime. The obtained limit-cycle families remain close to the nonlinear modal manifolds, and stable oscillations can still be generated under both stiffness variation and equilibrium adjustment. Energy analysis further shows that the input energy is mainly used to compensate for damping dissipation, indicating high energy efficiency. Experimental results further confirm that the proposed approach can induce robust self-excited limit-cycle oscillations in the tensegrity robotic leg. The proposed framework provides a systematic and physically interpretable approach for predicting and regulating self-excited oscillations in nonlinear tensegrity systems.