Non-prehensile Robotic Transportation of Liquid: MPC vs Time-Optimal Trajectory Planning

Robotic transport of liquid-filled containers without grasping--- the ``Waiter's Problem'' ---requires simultaneously preventing the object from sliding off the tray and the liquid from spilling over the rim. This thesis develops a unified optimal control framework that couples non-prehensile contact stability constraints with a spatial-pendulum sloshing model to address both challenges jointly. Two complementary strategies are proposed. The first is an offline time-optimal trajectory generator that parameterizes the full 6D Cartesian path (translation and orientation) with clamped cubic B-splines. Unlike prior approaches restricted to translation plus yaw, the optimizer jointly shapes position and orientation within geometric envelope constraints, exploiting tilting compensation for sloshing suppression. The resulting nonlinear program is solved via multiple shooting with an interior-point method, yielding minimum-time rest-to-rest trajectories for multi-container transport. The second strategy is a real-time kinematic Model Predictive Controller. The controller models the robot as a triple integrator with joint jerk as input, coupled to the liquid dynamics through forward kinematics, and employs a Kalman filter to reconstruct the unmeasured joint acceleration. A constraint dominance analysis proves that, under typical container geometries, the non-sliding friction-cone constraint subsumes both tipping and torsional slip, reducing the online constraint set to a single linearized Coulomb cone per container. Experimental validation on a UR5e robot confirms that the MPC reduces container displacement to sub-millimetre levels, keeps sloshing well below the spillage threshold, and satisfies all constraints across multiple trajectory types. Both approaches rely on the PEN6D spatial-pendulum model for sloshing prediction, extended to arbitrary 6D container motion and experimentally validated against camera-based surface measurements.