Time-Optimal Transport of Loosely Placed Liquid Filled Cups along Prescribed Paths

AI summary

Overview

Research area: Robotics — specifically optimal trajectory planning and control for industrial robot manipulators, intersecting with sloshing dynamics of liquids.

Technical level: Advanced. The paper is written for readers comfortable with optimal control formulations, Lagrangian mechanics, contact wrenches, and numerical optimization methods.

Scope: The paper formulates and numerically solves a time-optimal path-following optimal control problem (OCP) for a six-axis industrial robot transporting a loosely placed, liquid-filled cup along a prescribed lemniscate path without lifting, slipping, tipping, or spilling.

What This Paper Is About

A "waiter motion problem" asks a robot to carry a loosely placed object — here a liquid-filled cup sitting unrestrained on a tray mounted to the end effector — along a predefined geometric path. The goal is to move along that path in the shortest time while keeping the cup from sliding, lifting, or tipping, and keeping the liquid from sloshing over the rim. The authors build all of these task-specific constraints into a single optimal control problem and solve it numerically.

Key Contributions

1. A unified OCP formulation combining robot kinematics, rigid-body contact constraints, and fluid dynamics. The paper starts from the standard time-optimal path-following problem, then progressively adds dynamics, contact-wrench-based task constraints, and internal liquid dynamics into one optimization problem.

2. Explicit treatment of the loosely placed object via contact wrench constraints. Non-lifting, non-slipping, and non-tipping conditions are derived from the contact force and torque components and imposed as inequality constraints rather than assumed away.

3. A spherical-pendulum model of the liquid sloshing. The fluid is reduced to a lumped mass on a pendulum of length L with viscous damping d, attached to a frame aligned with the end-effector frame; the liquid surface is assumed to stay normal to the pendulum, and the pendulum's angles and angular velocities are bounded.

4. A numerical demonstration on a prescribed lemniscate path. Using direct multiple shooting in CasADi with IPOPT, the authors show a solution in which all robot joint limits, cup-stability constraints, and liquid constraints are satisfied.

Main Findings

• All robot axis limits were respected. Figure 3 shows that the transport task can be carried out without any violation of the velocity and acceleration constraints on the six robot axes (q_x, q_y, q_z, q_A, q_B, q_C), with velocities and accelerations shown normalized to their respective limits.

• All three cup-stability constraints remained satisfied. Figure 4 plots the non-lifting condition f_z ≥ 0, the non-slipping condition ||f_T|| ≤ μ_0 f_z, and the non-tipping condition sqrt(M_x² + M_y²) ≤ r_o f_z in the form (·) ≤ 0, and none is violated over the motion.

• The liquid pendulum states stayed within bounds. Figure 5 shows that the pendulum angles q_F = (φ, ϑ) and their velocities q̇_F complied with the box constraint x̄_F ≤ x_F ≤ x̄_F, where the limits used were q̄_F = π/18 rad and q̇̄_F = 5π/9 rad/s.

• Feasibility relies on model assumptions. The authors state that the liquid transport "can be assumed" to be carried out as presented, provided the assumed parameters are accurate and the liquid model is sufficiently precise. No physical robot experiment is reported — real-robot tests are described as planned.

• Problem parameters. The tray/cup geometry is r_AE,x = 0.07 m, r_AE,y = 0 m, r_AE,z = 0.055 m; the static friction coefficient is μ_0 = 0.35; the cup radius is r_o = 0.05 m; the fluid pendulum length is L = 0.027 m, the lumped mass is m = 0.55 kg, and viscous damping is d = 0.2 kg/s.

• Numerical setup. The OCP was solved with direct multiple shooting in the CasADi framework using IPOPT, with N = 400 shooting intervals, a 4th-order Runge-Kutta scheme for numerical integration of the cost and the ODE, and weighting factors w_t = 1 and w_u = 10⁻⁴.

• Optimized transport time. The resulting numerical value of the end time t_E is not reported in the provided paper content; the results are presented graphically as time evolutions over the trajectory.

Methodology in Plain English

The robot is a three-axis linear gantry extended by a three-axis rotational unit, giving six degrees of freedom. The point where the tray attaches is guided along a prescribed path described by a single scalar parameter σ that runs from 0 to 1; the optimization decides how fast that parameter advances in time.

Because the linear axes have no significant arm lengths, the authors limit joint accelerations instead of motor torques, which simplifies the dynamic model. For the cup resting on the tray, they write the equation of motion of the object as a separate subsystem and retain the contact force and torque between cup and tray. Those force and torque components translate directly into the physical conditions for not lifting off, not sliding (bounded by static friction μ_0), and not tipping over (bounded by cup radius r_o).

The liquid is not simulated as a full fluid. Instead it is replaced by a spherical pendulum of length L with a lumped mass m and viscous damping d, derived via Lagrange's equations of the second kind. The angle of this pendulum represents the tilt of the liquid surface, so simply bounding the pendulum angles and angular velocities keeps the liquid away from the cup's edge.

Finally, the whole thing becomes one OCP whose state includes σ, its first two time derivatives, the rotational joint states, and the fluid pendulum state, and whose input is the third derivative of σ plus the third derivatives of the rotational joints (i.e., jerk is piecewise constant, which yields continuous accelerations and torques). This is solved numerically with direct multiple shooting in CasADi/IPOPT on a lemniscate path in the horizontal plane.

Why This Matters

The work shows how task-specific physical constraints — contact conditions and fluid behavior — can be folded directly into the trajectory optimization rather than handled by a separate slow-down heuristic or by simply moving more cautiously. It is a step toward robots that can handle unstable, deformable, or liquid payloads at speed.

Real-world applications:

• Food and beverage service: a robot waiter carrying open cups or glasses on a tray in restaurants, cafeterias, or hotels.
• Logistics and intralogistics: transporting containers of liquid, such as chemical or food-industry vessels, on loosely placed trays or pallets.
• Laboratory and medical automation: moving open sample containers or reagent vessels where spillage would be hazardous or contaminate results.
• Manufacturing/handling: fast pick-and-place of containers and open-topped parts where the payload is not rigidly clamped to the gripper.

Industry relevance: The constraint-based formulation uses standard parameters (static friction coefficient measurable by a simple experiment, cup radius, pendulum length and damping) and standard optimization tooling (CasADi, IPOPT), making it directly transferable to industrial manipulator trajectory planners. The funding by the "LCM – K2 Center for Symbiotic Mechatronics" under the Austrian COMET-K2 program indicates an industry-oriented research context.

Future Directions

1. Experimental validation on real robots. The authors explicitly state that tests on real robots are planned to verify the procedure; none are reported in this paper.

2. Modeling the shift in the cup's center of gravity. The authors plan to extend the liquid-filled cup model so that the movement of the liquid, and the resulting change in center of gravity, is taken into account when computing the constraining forces.

3. Sensitivity to model accuracy. Because the conclusions are conditional on the assumed parameters being accurate and the pendulum model being sufficiently precise, quantifying how much sloshing-prediction error the spherical-pendulum approximation introduces would be a natural follow-up.

4. Generalization beyond the studied case. The results presented here are for a single lemniscate path with one fixed parameter set; extending the approach to other path geometries, cup sizes, fill levels, and liquid properties remains open.

Target Audience

Researchers and graduate students in robotics working on trajectory optimization, time-optimal path following, or non-prehensile manipulation; control engineers in industrial automation who need to move unstable or liquid payloads quickly; and anyone interested in combining rigid-body contact constraints with simplified fluid models inside a single optimal control problem. Readers without a background in optimal control, Lagrangian dynamics, or numerical shooting methods will find the formulation section demanding.

Authors’ abstract

Handling loosely placed objects with robotic manipulators is a difficult task from the point of view of trajectory planning and control. This becomes even more challenging when the object to be handled is a container filled with liquid. This paper addresses the task of transporting a liquid-filled cup placed on a tray along a prescribed path in shortest time. The objective is to minimize swapping, thus avoiding spillage of the fluid. To this end, the sloshing dynamics is incorporated into the dynamic model used within the optimal control problem formulation. The optimization problem is solved using a direct multiple shooting approach.