A Variable Centre-of-Mass Reduced-Order Model and Model-Predictive Control Framework for Stable Bipedal Walking

Stable locomotion of underactuated bipedal robots remains an open problem because of strong nonlinearity, hybrid contact dynamics and the inevitable mismatch between the modelled and physical centre of mass (CoM). This paper proposes an integrated trajectory-optimisation and model-predictive control (MPC) framework built around a reduced-order single-rigid-body dynamics (SRBD) representation augmented with a variable centre-of-mass coordinate frame. The new frame introduces a tunable offset between the geometric body frame and the true inertial centre, enabling online compensation of payload changes, manufacturing tolerances and modelling error without altering the kinematic description of the platform. The Newton-Euler equations are discretised in a fixed-step formulation and embedded into a finite-horizon quadratic programme that simultaneously regulates body attitude and linear and angular CoM motion while honouring friction-cone and ground-reaction limits at both feet. The optimal contact wrench is then redistributed to the actuated joints by a whole-body mapper. The framework is validated in three MuJoCo simulation campaigns on a 130 cm, 28 kg, ten-degree-of-freedom biped. In-place stepping yields physically consistent vertical reaction forces (peak approximately 235 N, matching the platform's static weight); forward velocity tracking settles within about two seconds for plus/minus 0.1 m/s step changes; and a deliberate 9 cm CoM offset that would otherwise produce approximately 0.025 rad of pitch bias and approximately 0.05 m/s of forward drift is fully suppressed by the proposed compensation. The results indicate that pairing a reduced-order model with an explicit CoM-bias correction yields a numerically lightweight yet robust controller well suited to onboard deployment on small-scale humanoid platforms.