Phase-dependent nonlinear MPC for stair climbing: Edge-pivot control with a double-pendulum model

This paper develops and validates optimal control for a double-pendulum and a hybrid switching controller for a wheel-based stair-climbing device (SCD). An energy-based linear and nonlinear model is derived and used to design LQR, observer-based state feedback, linear MPC, and nonlinear MPC (NMPC). The resulting nonlinear optimal control problems (OCPs), defined as nonlinear programs (NLPs), are solved with a numerical solver using a Real-Time Iteration (RTI) scheme, allowing NMPC to enforce the full nonlinear dynamics and state/input constraints. Under identical tests, the implemented NMPC best drives both links to upright equilibrium with the lowest tracking error for comparable effort. The implemented NMPC is then embedded in a phase-dependent switch where the LQR governs the nominal rolling, while at the critical edge pivot modeled as an inverted double-pendulum triggers the NMPC via hysteretic distance guards with contact confirmation and dwell time; after capture, the controller returns to LQR. This yields reliable, chatter-free switching and improved ascent/descent performance while saving energy.