A Solution to Slosh-free Robot Trajectory Optimization Rafael I. Cabral Muchacho Riddhiman Laha Luis F.C. Figueredo and Sami Haddadin Abstract This paper is about fast slosh-free ﬂuid transporta-

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A Solution to Slosh-free Robot Trajectory Optimization

Rafael I. Cabral Muchacho, Riddhiman Laha, Luis F.C. Figueredo, and Sami Haddadin

Abstract— This paper is about fast slosh-free ﬂuid transporta-

tion. Existing approaches are either computationally heavy or

only suitable for speciﬁc robots and container shapes. We model

the end effector as a point mass suspended by a spherical

pendulum and study the requirements for slosh-free motion and

the validity of the point mass model. In this approach, slosh-

free trajectories are generated by controlling the pendulum’s

pivot and simulating the motion of the point mass. We cast the

trajectory optimization problem as a quadratic program—this

strategy can be used to obtain valid control inputs. Through

simulations and experiments on a 7 DoF Franka Emika Panda

robot we validate the effectiveness of the proposed approach.

I. INTRODUCTION

In this work, we are interested in the problem of trans-

portation and optimal-manipulation of dynamic ﬂuids and

fragile materials. Particularly, we are focused on extreme

time-optimal solutions that leverage robot capabilities to

exceed human-level performance. Take for instance, the task

of transporting a full cup of coffee, a glass of wine, or a

hazardous liquid in industry. Although basic inspiration for

designing such a system can be gained from innate human

skills to carefully adjust the frequency of the containing

cup/container to match that of the unrestrained free surface,

achieving this in aggressive maneuvers for humans can be

extremely hard. Safety is the number one priority in this

case, as no one likes to spill a coffee—neither do robots.

Solutions often lie in minimizing higher derivative terms

such as jerk and snap of the trajectories through optimal

control strategies. Taking inspiration from a market solution

for coffee/liquid transportation as shown in Fig. 2—namely

the Spillnot mechanism [1], [2]—we propose an optimization-

based motion generation solution that provides above-human

real-time ﬂuid transfer at high speeds that predicts and

compensates reaction forces through a simpliﬁed closed-form

pendular-like dynamics—and thus complete the task without

spilling liquids even for aggressive motion proﬁles.

The problem of dynamic liquid behavior is well studied

and dates back to Navier, Stokes [3], the ﬂuid packaging

industry, and even the aerospace industry [4]. Motion induced

sloshing is also a classic problem in control theory [5],

[6] wherein the main idea is to study the sloshing mode

excited by the oscillations through resonance and additional

frequency analysis [7]. Among the robotics community, while

The authors are with Munich Institute of Robotics & Machine Intelli-

gence, Technische Universit

at M

unchen (TUM), Germany. This work was

funded by the Lighthouse Initiative Geriatronics by StMWi Bayern (Project X,

grant 5140951), LongLeif GaPa gGmbH (Project Y, grant 5140953), “Centre

for Tactile Internet with Human-in-the-Loop” (CeTI, grant 390696704) and

KI.FABRIK Bayern (grant DIK0249). S. Haddadin has a potential conﬂict of

interest as shareholder of Franka Emika GmbH. Email:

{rafael.cabral,

riddhiman.laha, luis.figueredo, haddadin }@tum.de

Fig. 1. Inspiration for our problem. Picture courtesy: Science Factory and

SpillNot trademark [1], [2]. See further details of the functioning of the the

SpillNot mechanism here.

a plethora of approaches have been proposed to address the

aforementioned transportation problem, most of them rely

on motion optimization and/or smoothness to minimize jerks

and acceleration leading to liquid sloshing, see for instance,

[8], [9]. A scarce amount of papers, have also addressed

the problem through ﬂuid analysis and optimization towards

trajectory smoothing and container tilt-coupling, i.e., coupling

with the trajectory [10], [11]. Such approaches however often

lead to computationally heavy, narrow and dedicated case-

speciﬁc solutions. Overall, existing frameworks addressing the

the posed problem are either suboptimal, or computationally

demanding in terms of solving the inverse motion problem.

Existing human-centred applications call for optimal, safe

and mostly real-time solutions, that make use of existing

collaborative robots capabilities and inner control-loops

running, for instance, at 1 KHz. Neglecting or overseeing the

robot’s capabilities hinders real-world applications where it

is desirable to have a faster and more effective solution.

Contrary to existing methods, this work instead adopts a

fundamentally different approach, and proposes solving the

spilling-free ﬂuid transportation problem by formulating it

as a linearized point mass system. Casting the problem like

this enables us to integrate the tilt of the container with the

arXiv:2210.12614v1 [cs.RO] 23 Oct 2022

Pivot

Reference

Trajectory

Virtual

Rod

Mass

End Effector

Path

Fig. 2. Simpliﬁed dynamics model that we use for designing the optimization

problem. The virtual pendulum forms the basis for analyzing the equations

of motion. Constrained robot end-effector motion is achieved by controlling

the pivot on the reference trajectory.

maximum possible task-space acceleration and jerk given

by the system constraints. It also allows designing optimal

motion generation through real-time quadratic programming

and therefore is able to run within the robot’s fast inner

control loop for real-time applications. To the best of our

knowledge, this is the ﬁrst work to propose a slosh-free

manipulation solution based on a closed-form linearized yet

accurate dynamics that account for translation and tilt-coupled

dynamics. Overall, the proposed method allows for real-time

capable motion generation for aggressive ﬂuid maneuvers

along a given path. A set of experiments exploring the Panda

robot arm capabilities are devised to highlight the efﬁciency

of the proposed method.

II. RELATED WORK

In the literature for ﬂuid dynamics planning and control,

researchers often estimate the interacting ﬂuid states by

deploying complex methods such as computational ﬂuid

dynamics (CFD) [12], [13]. However, CFD simulators for

even simple objects like coffee in a cup are usually computa-

tionally demanding and require hours to converge—deeming

them infeasible for most planning and control applications in

robotics. Numerical approaches for slosh modeling using

a simpliﬁed version of the Navier-Stokes equations are

usually decoupled in the sense that they involve an intensive

study of the properties of the system and an analysis of its

controllability [14]–[17]. The planar models [18], [19] are

limited to modeling the s-velocity proﬁles for the underlying

trajectory and designing time and frequency responses of the

containing liquid; these solutions do not scale up to high DoF

systems.

As an alternative solution, researchers have also tried to

model the system using approaches from Learning from

Demonstration (LfD) [20], [21]. The authors in [22] take an

optimization approach to pouring, based on a simpliﬁed ﬂuid

dynamics model.

In the realm of ﬂuid manipulation, the two most important

challenges are slosh-control [19], [23] and pouring [24], [25].

Therefore in this work, we exploit the capability of the robotic

arms to change the orientation of the container similar to the

works in [26]–[28]. The basic idea in these strategies is two

fold: (a) apply smoothing methods [10], [11] to the desired

trajectory, and (b) compensate for the expected remaining

oscillations of the liquid by varying the container’s orientation.

Furthermore, the novelty of our approach lies in the fact that

we do not require any further knowledge about the robot

except the provided reference trajectory and its kinematic

limits.

In this context, it is worth mentioning the works [29],

[30]. In [29], the proposed method explores the usage of

evolutionary optimization methods to compute spill-free

trajectories and impressively demonstrates the solution by

transporting a ﬁlled spoon. The evaluation step of the

algorithm is calculated through a detailed CFD simulation,

which is accompanied by high computation expenses. The

work in [30] relies on direct and active compensation to

investigate a 2D active control strategy to avoid spillage,

sloshing and more generally to ensure a safe transport of

delicate objects. To implement the control strategy they mount

a parallel manipulator on the vehicle. The object is then

carried by the manipulator and its motion “emulates a free

swinging pendulum to minimize lateral forces acting on the

object”. Regarding the length of the pendulum, instead of

using a ﬁxed value, they take it as a control variable. The

modeling of pendulum-like dynamics for the motion of liquids

is also addressed at [31], which proposes human grasping

strategies to better compensate oscillations and resonance of

the modelled dynamics. A similar model can be used for

the analysis of human motion through dynamic primitives;

the authors of [32] compare the human motion to rest-to-

rest control strategies for the unconstrained manipulation of

non-rigid objects.

In this work, we also take inspiration from pendulum-like

dynamics, yet differently than existing literature we focus

on the linear approximation of the 3D pendulum to deﬁne

the trajectory optimization problem as a quadratic-program,

directly combining the smoothing and tilt coupling steps and

allowing for the enforcement of additional constraints.

III. PRELIMINARIES AND CONCEPT PROPOSAL

This section presents the underlying intuition behind our

approach, mathematical background and the fundamentals of

the proposed motion generation and control problems.

A. Conditions for Slosh-free Motion

Intuitively, one can reason about the sloshing problem

as being caused by the reaction forces exerted on the

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ASolutiontoSlosh-freeRobotTrajectoryOptimizationRafaelI.CabralMuchacho,RiddhimanLaha,LuisF.C.Figueredo,andSamiHaddadinAbstractThispaperisaboutfastslosh-freeuidtransporta-tion.Existingapproachesareeithercomputationallyheavyoronlysuitableforspecicrobotsandcontainershapes.Wemodeltheendeffectorasapoi...

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A Solution to Slosh-free Robot Trajectory Optimization Rafael I. Cabral Muchacho Riddhiman Laha Luis F.C. Figueredo and Sami Haddadin Abstract This paper is about fast slosh-free ﬂuid transporta-

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