Minimum-time anti-sloshing motion planning for an open axisymmetric container transported by a 3-DOF planar motor

We consider the problem of transporting an open liquid-filled container along a straight line in minimum time while preventing liquid spilling. The container is transported by a planar motor with three degrees of freedom: horizontal translation, vertical translation, and pitching angle. Under the assumptions of linear sloshing, small liquid oscillations, and an axisymmetric container, the liquid dynamics are represented by an equivalent damped pendulum. The resulting minimum-time motion planning problem is non-convex. To obtain a convex formulation, the nonlinear terms are approximated using first-order Taylor expansions and McCormick relaxations. A general proportional relation between the vertical displacement of the liquid free surface and the pendulum angle is also derived for axisymmetric containers. The final time is fixed and the minimum feasible time is found through a bisection procedure, allowing the global optimum of the approximated problem to be obtained. Experiments using different axisymmetric container shapes and sloshing limits demonstrate the effectiveness of the approach. The approximation error is evaluated by measuring the constraint violation of the approximated solution with respect to the original non-convex problem, showing small violations in all considered cases. The contribution of vertical translation and pitching is also evaluated, showing that vertical translation has a negligible impact on the optimal transport time, while pitching can provide significant time reductions, especially for low sloshing limits, up to 76.26%. Finally, the effect of container geometry is investigated, showing that the minimum transport time is little sensitive to geometry changes, while the optimal control depends on the geometry.