Reachability Guarantees for Cart-Pole Swing-Up and Stabilization

Abstract

The cart-pole swing-up is a canonical benchmark for nonlinear control of underactuated systems, yet an end-to-end guarantee linking the global swing-up maneuver to the local stabilizer is seldom formalized. We present a reachability analysis of a switched energy-based/LQR controller that certifies convergence to the upright equilibrium from a compact set of initial conditions. The swing-up design exploits the phase-space geometry of the conservative pendulum: the upright equilibrium lies on the homoclinic orbit, and an energy-shaping law drives the energy error to zero, steering the pendulum onto this orbit; convergence follows from LaSalle's invariance principle. An augmented Lyapunov function additionally regulates the steady-state cart velocity to zero, and we prove almost-global convergence of the resulting closed-loop system. A local LQR with a certified ellipsoidal region of attraction stabilizes the upright equilibrium, and we verify numerically that the swing-up phase delivers the state into this region, formalizing the handoff. Numerical simulations corroborate the theoretical analysis.