Lyapunov-Certified Parameter Search and Terrain-Adaptive Feedback Control for Legged Locomotion Systems

Abstract

: Stable high-speed locomotion of legged robotic platforms depends on gait parameters that are usually tuned by hand, with no guarantee that the resulting cycle is stable or efficient. This paper builds a hybrid dynamical model of the center-of-mass trajectory using a spring-loaded inverted pendulum abstraction, with separate governing equations for the contact and flight phases and explicit event conditions for phase switching. An adaptive-step Runge-Kutta integrator recovers the full single-cycle trajectory. Stability is certified by the largest Lyapunov exponent rather than by visual inspection of the trajectory. A global grid search over contact durations from 0.05 to 0.35 seconds and flight durations from 0.10 to 0.35 seconds, at a fixed forward speed of 3 meters per second, isolates the feasible region in which periodic motion remains stable. Within that region, minimizing mechanical energy per unit distance selects 0.2470 seconds of contact and 0.1293 seconds of flight, at a cost of 45.12 joules per meter. The model is then extended with a terrain height function, and a leg-length feedback law combined with a feedforward and proportional-integral foothold controller keeps the largest exponent negative across three consecutive cycles on uneven ground. The pipeline turns gait tuning into a certified search problem.