Model Predictive Control for Non-Prehensile Robotic Manipulation of Liquid Containers with Sloshing Boundedness Guarantees

Transporting open liquid containers on a robotic tray without grasping requires simultaneously guaranteeing contact stability and spill prevention, two objectives inherently coupled through the common end-effector acceleration. This paper presents a real-time Model Predictive Control (MPC) framework that embeds both requirements within a single optimization. The robot is modeled as a jerk-controlled triple integrator, while liquid dynamics are captured by a nonlinear equivalent spherical-pendulum model under general 6D end-effector motion. The formulation supports an arbitrary number of containers, each with independent sloshing dynamics and contact constraints, accounting for coupled translational and rotational effects — including tangential and centripetal contributions — and providing a closed-form estimate of the free-surface elevation enforced as a hard constraint to limit spillage risk. Non-prehensile stability is ensured through linearized Coulomb friction-cone constraints that prevent both detachment and sliding. The resulting nonlinear program is solved in real time via an SQP-based solver within a multi-rate architecture bridging a 50 Hz MPC planner and a 500 Hz robot velocity interface. Experiments on a UR5e manipulator across three trajectories confirm that the controller maintains stable contact and significantly reduces sloshing compared to unconstrained execution, consistently respecting the prescribed wave-height bound.