Energy-consistent linearization of hybrid limit-cycle systems via expansion-point design

Linearization of hybrid limit-cycle systems is traditionally performed around unstable equilibrium points of the continuous dynamics. However, in periodic locomotion systems, such equilibria do not constitute invariant structures of the cycle and therefore do not necessarily preserve its essential discrete behavior. In this paper, we reformulate linearization from a cycle-level perspective by treating the expansion point of the Taylor approximation as a design parameter. Instead of enforcing local consistency at an unstable equilibrium, the expansion point is determined such that the restored mechanical energy and the step-to-step kinetic energy recurrence of the linearized model match those of the nonlinear system. The proposed framework is demonstrated using the passive rimless wheel, a canonical hybrid limit-cycle system. By generalizing the expansion point of the gravitational term, the discrete energy evolution of the linearized model becomes identical to that of the nonlinear model while preserving analytical tractability. An approximate analytical expression for the steady step period is derived, and numerical simulations confirm its high accuracy over a wide range of slope angles. The results establish a new perspective on linearization in hybrid periodic systems as a cycle-consistent design problem rather than a local equilibrium-based approximation.