Control of Liquid Handling Robotic Systems:a Feed-Forward Approach to Suppress Sloshing

Lorenzo Moriello, Luigi Biagiotti, Claudio Melchiorri, Andrea Paoli

Abstract— This paper presents a feed-forward approach toreduce sloshing dynamics in liquid handling robotic systems.According to our solution, the dynamics of a liquid into anopen vessel manipulated by a robot can be described by meansof a spherical pendulum mechanical model. By doing this, thesloshing problem can be addressed as a vibration suppressionproblem for a second order system. More in details, thependulum model is utilized to tune an exponential filter whichshapes the reference trajectory for the robot, thus achieving asloshing-free motion of the liquid inside the vessel.

I. I NTRODUCTION

The current industrial scenario is becoming more and morecompetitive and, as a consequence, modern manufacturingis always more strict with regard to performances anddependability. In this respect, industrial automation is playinga key role as a powerful means to optimize manufactur-ing processes and maximize their efficiency and flexibilityof production lines. For this reason, industrial robotics ispervading many diverse industrial fields to overcome thesevere and unacceptable limitations that are due to the useof rigid automation systems. A particularly interesting andpromising case is the use of industrial robots in the foodprocessing industry to increase flexibility, performancesandsystem dependability. In this framework, not only robots areutilized to perform standard heavy-duty tasks, but also toaccomplish tasks which are traditionally executed by humanoperators or by means of process control systems (see [1]).

The innovative use of industrial robots in non-traditionalindustrial sectors has drawn the attention of scientists andtechnologists to novel challenging manipulation tasks as theliquid handling one. In this regard, manipulating and trans-porting liquids is the order of the day in the food industry andthe use of robots to solve this problem is acquiring an ever-increasing importance in the sector. Similarly, many otherindustrial sectors could benefit from an effective solutionto this problem. Among others it worths mention the steelindustry and the problem of handling melted metal by meansof industrial robots (see [2]and [3]).

Generally speaking, the problem of using robots to manip-ulate a vessel filled with liquid is an extremely complex andchallenging task. Indeed, this is due to the complex nonlinear

L. Moriello and C. Melchiorri are with the Department of Electrical,Electronic and Information Engineering “Guglielmo Marconi”, Universityof Bologna, Viale del Risorgimento 2, 40136 Bologna, Italy,e-mail:{lorenzo.moriello2},{claudio.melchiorri}@unibo.it.

L. Biagiotti is with the Department of Engineering “Enzo Ferrari”,University of Modena and Reggio Emilia, via Pietro Vivarelli 10, 41125Modena, Italy, e-mail: luigi.biagiotti@unimore.it.

A. Paoli is with the School of Engineering, University of Lincoln,Brayford Pool, LN6 7TS, Lincoln, UK, e-mail: APaoli@lincoln.ac.uk.

fluid dynamics that arise within the vessel. In engineeringliterature many approaches are proposed to mathematicallydescribe these dynamics (see [4] and [5]).

With respect to the industrial automation sector, manysolutions have been proposed to achieve fast motion (typ-ically in conveyors or within automatic machineries) whileavoid sloshing flows. All these methodologies rely on theassumption of a simplified linear model of the sloshingdynamics. In [6], [9] the sloshing flow is treated as a model-based disturbance suppression problem that can be solved bymeans of proper feedback control strategies, while in [7], [8]feed-forward methods (i.e., generating a proper referencetra-jectory for the machine) are proposed. Note that feed-forwardapproaches may be preferable in industrial applicationssince feedback-based solutions require the implementationof specific control architectures, which cannot be realizedon standard robots. In fact, factory controllers equippingcommercial manipulators are generally unaccessible to theuser, and the only way to interact with them is providing aproper reference input.

In this paper a feed-forward approach to reduce sloshingdynamics in liquid handling robotic systems is presentedand experimentally tested. Specifically, sloshing flows aremodeled as a spherical pendulum model depending on thevessel geometry and the property of the liquid (see Sec. II).In Sec. III the problem is then addressed as a vibrationsuppression problem and the reference trajectory for therobot is generated utilizing a properly designed exponentialfilter [10]. The filter is finally implemented as a discreteFIR filter whose performances and robustness properties arepresented and discussed. To conclude, Sec. IV presents arange of experimental results that prove the validity andeffectiveness of the method.

II. M ODELLING THE SLOSHING EFFECT

Modelling the dynamics of a fluid inside a moving con-tainer can be very sophisticated. A rigorous mathematicalmodelling of slosh dynamics employs the Navier-Stokesequations, a set of nonlinear, partial differential equationswhich take into account many characteristic parameters ofboth the fluid and the vessel geometry.However, if the oscillations are confined (i.e., nonlinearsloshing phenomena are avoided), it is possible to describethe liquid motion by means of an equivalent (linear) mechan-ical model that takes into account two distinct phenomena,see Fig. 1(a): the former is the motion of the fluid directlyproportional to the acceleration of the rigid container which

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Fig. 1. Linear pendulum based model (a) and spherical pendulum basedmodel (b) of liquid sloshing.

is described by the massm0, while the latter is the superpo-sition of j different sloshing modes described by means ofa series of simple pendulums, linearized about the verticalposition [4].Note that it is common practice for sloshing control purposes,to take into account only the first asymmetric mode (lowestfrequency), i.e. the motion of the free-surface is describedby means of a single pendulum. In this case the liquidsurface can be considered as a planar surface which is alwaysorthogonal to the pendulum itself as in Fig. 1(b).Mechanical parameters of the pendulum can be directlyobtained from the linearization of Navier-Stokes equationsand the typical relationωn1 =

√

g/l1, wherel1 is the lengthof the pendulum.In case of a fluid in an upright cylindrical container, thenatural frequency of thej-th sloshing mode results

ωnj =

√

g ξjR

tanh

(

h ξjR

)

, (1)

whereg is the gravity constant,R is the cylinder radius,his the liquid height, andξj is the j-th root of the derivativeof Bessel function of the first kind.In addition, the damping coefficientδj of the generic sloshingmodej can be deduced from the logarithmic decay ratio ofthe slosh due to the energy dissipation on the free surface, onthe sidewalls and on the tank bottom. However, when onlythe first sloshing mode is considered, the damping ratio isusually defined by means of an empirical relationship whichtakes into account the kinematic viscosityν of the fluid andthe vessel parametersh andR (see [5])

δ1 =2.89

π

√

ν√

R3g

[

1 +0.318

sinh(1.84hR

)

1− hR

cosh(1.84hR

)

]

. (2)

When a three dimensional motion of the container is takeninto account, the pendulum model can be extended byconsidering a spherical pendulum as in Fig. 1(b). By doingthis, it is also possible to describe particular dynamic featureslike rotary sloshing.

III. E XPONENTIAL FILTER FOR SLOSHING SUPPRESSION

Controlling the motion of a vessel filled with liquid whileavoiding sloshing can be extremely challenging using a tra-ditional feedback control architecture. The main problem is

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that the control loop should instantly react to the high speeddynamics of the fluid and this would require an extremelyfast and accurate set of sensors and a complex non-linearcontroller. Conversely, a feed-forward control architectureallows to achieve good results utilizing a simpler controland technological architecture.

Considering the mechanical pendulum model derived inSec. II and its linearization around the equilibrium pointcorresponding to a non-sloshing state, sloshing flows can bedescribed by means of the characteristic transfer functionofa linear second order system

G(s) =ω2n

s2 + 2δωns+ ω2n

(3)

beingδ the damping coefficient andωn the natural frequencyof the system.As a consequence, the problem of reducing the sloshingeffect can be approached as a typical vibration suppressionproblem. In literature many ad hoc solutions are proposedto face this problem, since simple workarounds such asjerk limitation or acceleration smoothing doesn’t guaranteevibration suppression, especially for damped vibrations.In[10] authors present a simple and effective method to reducethe oscillation of a vibratory servo system applying a properfeed-forward control action able to properly filter the givenreference signal. Considering a dynamic system with oscil-lating behaviour described by (3) fed by a step input, theproposed exponential filter

Fexp(s) =α

eαTe − 1

1− eαTe e−Te s

s− α(4)

guarantees the complete residual vibration suppression if

α = −δ ωn , Te =2π

ωn

√1− δ2

. (5)

The result, shown in Fig. 2(b), can be explained by analyzingthe poles and zeros ofFexp(s) when the conditions (5) aremet. In this case, the filter has a real pole inp = −δ ωn,while the zeros, obtained from1−eαTe e−Te s = 0, arezn =−δ ωn±j n ωn

√1− δ2, n = 0, 1, . . ., see Fig. 3. Therefore,

Fexp(s) introduces an infinite number of zeros located alonga vertical line in the complex plane. In particular, note thatthe zeros obtained withn = 1 exactly cancel the poles ofthe plantG(s), thus allowing a perfect compensation of thevibrating dynamics. More in general the proposed method

Re{s}

Im{s}

−δωn

−j3ωn

√1− δ2

−j2ωn

√1− δ2

−jωn

√1− δ2

j3ωn

√1− δ2

j2ωn

√1− δ2

jωn

√1− δ2

Fig. 3. Pole-zero map ofFexp(s).

can be extended to any type of Single Input Single Output(SISO) Linear Time-Invariant (LTI) system, characterizedbyone or more oscillating modes. Specifically, given a dynamicsystem modelled as

G(s) =N(s)

D(s)(s2 + 2δωns+ ω2n)

whereN(s) andD(s) are generic polynomial (D(s) beingHurwitz), it is possible to show that the contribution to theresponse of the oscillating mode characterized by(δ, ωn) canbe completely nullifiedTe seconds after the application ofthe input signal by filtering this latter through a properlytuned exponential filterFexp(s).It is important to stress that the complete vibration suppres-sion requires the exact knowledge of the system’s param-eters, which is quite difficult to deduce in many practicalsituation. However, when a parameter mismatch occurs, thecancellation is only partial but, in any case, the residualvibration is significantly reduced. In particular, it has beenproved that the method based on exponential filters presentsan increased robustness compared to other common feed-forward techniques for vibration reduction such as InputShaping and System Dynamic Inversion.Also, it has to be noted that the capability of the presentedmethod of cancelling residual vibrations does not dependon the particular input considered. As a matter of fact,the resulting effect is a filtered trajectory which preservesthe desired steady states (due to the unitary static gain ofthe exponential filter) and is characterized be and increasedsmoothness (as the filter increases of one the number ofcontinuous derivatives of the trajectory).In the next section this methodology is applied to a roboticliquid handling system and its performances are discussed.

IV. EXPERIMENTAL RESULTS

In order to evaluate the proposed method the experimentalsetup shown in Fig. 4 has been arranged. The motionsystem is composed of a COMAU Smart5 Six industrialrobotic arm, a COMAU C4G Controller and a standard PCwith an Intel Core 2 Duo 2.4 GHz processor and 1 GB ofRAM. The COMAU Smart5 Six is a 6 DOF robot withanthropomorphic structure and a 6 Kg payload. The robotis driven by the COMAU C4G Controller that performs boththe position/velocity control (adaptive control) and the powerstage management with current control of each joint. TheC4G Controller also implements a software option called

Real Time PC

Position,

Velocity,

Current controlCu

COMAU

Smart5 SiX

COMAU C4G Controller

Trajectory

Generation

RGB-D camera

on,

ty,

control

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Smart5 S

D cameraRGB D

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stream

Vision PC

Liquid

Container

xy

z

Fig. 4. Experimental setup based on a Comau Smart5 Six industrialmanipulator.

Online TrajectoryGenerator

ExponentialFilter

InverseKinematics

Robot

s

qk : [x, y, z, r, p, y]

s

sJ , J = 1 . . . 6

WorkspaceFrame

Joint SpaceFrame

Fig. 5. Set-point generation algorithm for the industrial robot.

“C4G OPEN” that allows the integration of the robot controlunit with the external personal computer, in order to developcomplex control systems at high hierarchical level. The C4GOpen architecture is based on a real time communication onEthernet network between the controller and the real timePC. In particular the PC runs on the real-time operatingsystem RTAI-Linux on a Ubuntu NATTY distribution withLinux kernel 2.6.38.8 and RTAI 3.9 that allows the trajectorygenerator to run with a sampling periodTs = 2ms.A cylindrical steel pot of diameter200mm is rigidly attachedon the robot flange and a stiff structure supports an ASUSXtion PRO Live RGB-D camera able to monitor the liquidsurface and detect the depth information. The RGB-D camerais connected to a standard laptop running ROS operatingsystem; this subsystem is dedicated to video acquisition andoff-line elaboration. The vision system is able to acquire bothRGB and depth video streams at a frame rate of30 fps. Inall the experiments tap water is used to fill the container andacrylic blue dye is added to improve detection by the camera.It has to be noted that the camera is used for inspection andmeasurement of the resulting slosh only, without any purposeof introducing feedback control loops since the presentedapproach is totally feed-forward and model-based.For the implementation of the trajectory generator the Mat-lab/Simulink environment has been used. Since the robotcontroller requires the position set-point within the JointSpace Frame, the online generation algorithm depicted inFig. 5 has been implemented, where the exponential filterFexp(z) is realized by means of a discrete FIR filter, derivedby Z-transformingFexp(s) in (4) with a sampling timeTs.Filter’s parameters are defined accordingly toωn1 and δ1obtained from (1) and (2), using the geometrical features of

ωn1ωn1ωn1 [rad/s] δ1δ1δ1

Vol. [litres] h [m] Theor. Exper. Theor. Exper.

1 0.032 9.755 9.922 0.0053 0.0089

2 0.064 12.205 12.542 0.0037 0.0062

TABLE I

SYSTEM PARAMETERS WITHR = 0.1[m] AND ν = 1, 004−6[m2/s].

the pot and the characteristics of the water. It has to be notedthat the trajectory must be filtered in the workspace frame,since the vibratory system is defined in spatial coordinates.

A. Experimental validation

Firstly, in order to validate the sloshing model, a simplethird order point-to-point motion has been commanded to therobot carrying2 litres of water. In Fig. 6 in red is reported thebasic motion profile which is a Double-S velocity trajectoryalong the y axis of the workspace frame. The blue linecorresponds to the same trajectory filtered by means of theexponential filterFexp(z), whose characteristic parametersare defined according to (5) using the values reported inTable I, which refers to the model of a cylindrical containerof radiusR = 0.1[m] filled of water with kinematic viscosityν = 1, 004−6[m2/s]. In Fig. 6 are also highlighted thedifferent time duration of the two trajectories, in particularin blue is visible the delayTe introduced by the filter.In order to characterize the effectiveness of the proposedmethod, the depth measurement given by the RGB-D camerais analyzed to derive the motion of the liquid surface.The off-line algorithm developed using Matlab environmentfollows the considerations of the pendulum model in Sec. II.Since the liquid free-surface is assumed to be a plane, everyraw depth frame is processed in order to find the interpolatingplane z = ∆h+Kxx+Kyy, which best fits the depth pixelscorresponding to the water surface, assuming as origin of themeasurement frame the center of the free-surface in quietcondition. In this way any frame describes the instantaneousposition of the surface allowing to analyze the motion of theliquid when applying the trajectory. Moreover in order toclearly understand the results, the angles between the planeand the coordinate axes are reported in degrees in lieu of theangular coefficients

θx = arctan(Kx), θy = arctan(Ky),

as shown in Fig. 7.In Fig. 8 the comparison of the water slosh without and withthe exponential filter is reported. First of all by analyzingtheresponse to the non-filtered motion, it is possible to deducethe experimental sloshing parametersω⋆

n1, δ⋆1 by simpleidentification techniques. In particular, the resulting valuesω⋆n1 = 12.542[rad/s] andδ⋆1 = 0.0062 are comparable with

the theoretical parameters in Table I. The same test with 1litre of water has also been performed but is not shown herefor the sake of brevity. However in Table I the experimentalresults are reported demonstrating the validity of the model.The effect of the filtered trajectory on the liquid motion in

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Fig. 6. Third order trajectory set-point (red dashed line) and filteredtrajectory (blue line) for the model validation. The red andblue zonesindicate the time duration of the trajectories, respectively without and withthe exponential filter.

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Fig. 8 is greatly appreciable along the axis on which therobot motion occurs (θy(t)). In particular the resulting sloshwhen applying the filtered trajectory is reduced by the90%with respect to the non-filtered third order trajectory. On theother axis (θx(t)) instead, the result shows a similar behaviorbut in this case the slosh is due to the not exact synchronismbetween the robot joints when reaching the discrete set-pointsof the desired trajectory, causing a non perfectly rectilinearmotion.It is worth noting that, by definition, the effect of the sloshingsuppression has to be considered after the end of the motion.While the robot is moving the liquid into the container cannot

remain still due to its inertia. However, it can be noted thatthe motion of the surface in the non-filtered case experiencesa superposition of the inertial effect and the vibrating effect,while using the exponential filter the angular displacementof the surface with respect to the equilibrium is only due tothe inertia (i.e., to the acceleration applied to the water).Finally, it is important to note that the proposed algorithmforthe surface reconstruction introduces a term∆h which is notpresent in the classical pendulum model. Moreover in papersdealing with sloshing control, the translational effect ofthesurface plane is usually neglected because the resulting sloshin experimental activities is often measured by means of oneor more depth sensors, placed on the wall of the containers.Conversely, by means of a 3D reconstruction it is possible toappreciate that the liquid free-surface slightly translates dueto mass transfer.To conclude this section, we can state that experimentalresults demonstrate that the pendulum model properly de-scribes the rotational behavior of the surface, that is theanglesθy(t) andθx(t).

B. Implementation of a complex task

In order to test this method under realistic operationalconditions, a complex task composed of several motion tractsalternated with stops has been planned, to simulate a typicalindustrial task in which the robot is demanded to move theend-effector to several points in its workspace. To this aim, acubic B-Spline trajectory is adopted as basic motion profile.Splines and B-Splines are de-facto standard tools used in theindustrial field for planning complex motions interpolatinga set of given via-pointsq⋆i , i = 0, . . . , n − 1 at timeinstantsti. In particular uniform B-splines trajectories areconsidered, that is B-Splines characterized by an equally-spaced distribution of the knotsti (see [11]):ti+1 − ti =T, i = 0, . . . n − 2. In this case the online generation ofthe trajectory can be easily obtained by means of a chainof discrete FIR filters as described in [12] and [13]. Theexponential filter is then inserted in cascade configurationwith respect to the B-Spline generator, as shown by thealgorithm depicted in Fig. 5.In Fig. 9 the geometrical description of the planned B-Splinetrajectory is reported along the filtered trajectory. The greycircles highlight the positions in which the motion stops,therefore where the sloshing will be suppressed. Moreoverin Fig. 10(a) is reported the tangential acceleration of thereference trajectory where is visible the smoothing effectdueto the exponential filter (in blue). Also it can be noted thateven if the entire trajectory is composed of several tracts,the exponential filter introduces a total delay on the wholetrajectory of justTe seconds, which is0.644 seconds in the1 litre case and0.515 seconds in the 2 litres one.In Fig. 10(b) the comparison of the water slosh without andwith the exponential filter in the same condition of the valida-tion experiment (2 litres of water) is reported. It can be easilynoticed that the resulting slosh is greatly reduced, especiallywhile the robot stands in a stop period (grey zones). It is alsoworth noting that the sloshing suppression by the exponential

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Fig. 9. Geometrical description of the planned spline trajectory used in theexperimental test (in red) and filtered trajectory (in blue). Motion tracts areindicated with numbers and positions where the motion stopsare indicatedwith letters.

filter prevents the superposition of sloshing modes and iner-tial modes when the motion restart. Conversely, in the non-filtered experiment (red-dashed line) the total slosh duringmotion tracts increases due to the persistence of oscillationscaused by the previous movements.A similar behavior can be seen in Fig. 10(c) where thesame trajectory has been applied to the robot with thecontainer filled of just 1 litre of water. In this case theexponential filter has been implemented accordingly to themodel parameters considering 1 litre of volume (see Tab.I). The result confirms the validity of the model and theeffectiveness of the proposed method.In addition, the same experiment has been carried out apply-ing an accelerated trajectory to the robot carrying 1 litre ofwater. This means that the geometrical path of the trajectoryis the same as before but in this case the via-points of the B-Spline trajectory are interpolated with a knot span duration Treduced of the 25%. Thus velocity and acceleration increaseand, as a consequence, the liquid into the container is moreexcited. In Fig. 10(d) the effect of the sloshing suppression isclearly visible . In particular, despite the higher accelerationimpressed on the water which causes wider oscillation duringthe motion, it can be noted that the filtered trajectory greatlysuppresses the sloshing effect when the motion stops.Also is important to report that in this particular experimentthe non-filtered trajectory causes a huge sloshing behaviorwith liquid spilling out of the container. As mentioned inSec. II for high sloshing level the linearized model is nomore valid to describe the surface motion, therefore theresult in Fig. 10(d) for the non-filtered case (red dashed line)is a coarsest approximation. This is also confirmed by theconfidence intervals for the estimated plane coefficients ofthe interpolation algorithm, which are used to reconstructthe surface motion. In particular, in this case the confidenceinterval can be an order of magnitude greater than all theother tests.

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Fig. 10. Tangential acceleration of the B-Spline trajectory without (reddashed line) and with (blue line) exponential filter (a), andreconstructionof the water surface when applying the trajectory to the robot carrying 2litres (b) and 1 litre (c) of water. In (d) a speeded up B-Spline trajectory(red dashed line) and respective filtered trajectory (blue line) is applied tothe robot carrying1 litres of water. The gray zones indicate the motionstops.

V. CONCLUSIONS

A novel method to generate trajectories for sloshing-free motion of liquid filled vessels has been proposed andexperimentally tested. The method is based on a linearizedmechanical equivalent model of sloshing. This simplifiedspherical pendulum model is used to derive a set of parame-ters by means of which an exponential filter is designed andimplemented in cascade configuration to a generic trajectorygenerator. As proved by experimental tests, this simpleyet effective method allows a significant reduction of thesloshing effect during complex motion tasks and proves tobe very robust with respect to parametrical uncertainties.The presented method can be easily applied to many stan-dard industrial robotics systems without having to add anyexternal equipment or modify the control system of therobot. The exponential filter implemented as a FIR filter, canbe plugged after any online trajectory generator providingsloshing-free motion without altering kinematic constraintsof the desired trajectory. Also with respect to input shapingmethods, the exponential filter increases the number ofcontinuous derivatives which may simplify the generationof the desired trajectory.

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