research article integrated trajectory planning and...

Research ArticleIntegrated Trajectory Planning and SloshingSuppression for Three-Dimensional Motion ofLiquid Container Transfer Robot Arm

Wisnu Aribowo12 Takahito Yamashita1 and Kazuhiko Terashima1

1Department of Mechanical Engineering Toyohashi University of Technology Toyohashi 441-8580 Japan2Department of Industrial Engineering Bandung Institute of Technology Bandung 40132 Indonesia

Correspondence should be addressed to Wisnu Aribowo wisnumailtiitbacid

Received 17 November 2014 Accepted 14 April 2015

Academic Editor Rene V Mayorga

Copyright copy 2015 Wisnu Aribowo et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

For liquid transfer system in three-dimensional space the use of multijoint robot arm provides much flexibility To realize quickpoint-to-point motion with minimal sloshing in such system we propose an integrated framework of trajectory planning andsloshing suppression The robot motion is decomposed into translational motion of the robot wrist and rotational motion of therobot hand to ensure the upright orientation of the liquid containerThe trajectory planning for the translational motion is based oncubic spline optimization with free via points that produces smooth trajectory in joint space while it still allows obstacle avoidancein task space Input shaping technique is applied in the task space to suppress the motion induced sloshing which is modeled asspherical pendulum with moving support It has been found through simulations and experiments that the proposed approach iseffective in generating quick motion with low amount of sloshing

1 Introduction

In this research we address a case where a robot arm isused to move a liquid filled container in three-dimensionalspace Some example application cases for this setting aretransfer of molten metal in casting industries and liquidcontainers handling by service robots The use of articulatedrobot arm with high degree of freedom for these purposesprovides more dexterity and flexibility when compared withthe more dedicated type of liquid transfer devices Thereare two main performance characteristics to consider inthis system quick motion and minimal sloshing Those twocriteria are conflicting as quick motion tends to generatemuch sloshing of the contained liquid Sloshingmay cause theliquid to spill out of the container and in the case of moltenmetal too much sloshing may affect the physical propertiesof the liquid Moreover other task for example pouringcannot be performed when residual sloshing still occursthereby potentially reducing the system productivity Thusthe challenge in this problem is to generate trajectories withconcurrent consideration of motion time minimization and

sloshing suppression for point to point motion in a three-dimensional working space where static obstacles may exist

The motion speed of robot arm is mainly constrained byits kinematic and dynamic limit of the joints One method togenerate trajectories for such system is by posing the optimalcontrol problem as a nonlinear optimization problem wherethe system states are discretized in some way To reduce thelarge solution space and its associated computation timethe trajectory is often parameterized into a known structurefor example polynomial function Among the polynomialsplines cubic spline is the lowest order polynomial trajectorythat guarantees continuation in acceleration and velocity aswell as limited jerk Higher order polynomials would result insmoother profile but longer motion time while lower orderones would result in unlimited jerk

Numerous cubic spline algorithms and optimizationschemes have been proposed in the literature [1ndash5] Most ofthem try to optimize either the motion time or the locationof knots depending on the application needs Relatively fewaddressed the optimization of time and location of knotsconcurrently [5] In the latter approach the location of

Hindawi Publishing CorporationJournal of RoboticsVolume 2015 Article ID 279460 15 pageshttpdxdoiorg1011552015279460

2 Journal of Robotics

knots is not fixed which is suitable for trajectory planningproblem with obstacle avoidance In addition it often resultsin a quicker motion because there is less restriction for theoptimization algorithm to reach optimal velocity profile

The smooth cubic spline trajectories induce less vibrationwhen compared to nonsmooth or lower order trajectoriesHowever in cases where minimal sloshing is requiredsmooth trajectory alone is not enough and it needs to besupplemented with explicit sloshing control strategy Therehave been numerous researches dealing with sloshing sup-pression in liquid container transfer system Feedback controlbased approaches present good and relatively robust sloshingcontrol ability [6 7] However in some practical casessensor measurements cannot always be reliable or even aredifficult to perform for example in the case of controllingsloshing of high-temperature molten metal In these casesfeedforward approaches [8ndash14] are more useful Based onsloshing models the behavior of the system can be predictedand thus the trajectory or motion path can be designedaccordingly Examples include generalized bang-bang control[9] infinite impulse response (IIR) filter [8] input shapingfilter [10ndash13] andHybrid Shape Approach filters [14] Amongthem the input shaping approach is particularly interestingbecause of its simplicity and effectiveness in practice It worksby decomposing the command input signal into two ormoreusing only the knowledge of natural frequency and dampingratio of the system [15 16] Over time various type of inputshapers with different characteristics have been proposedand they have been successfully applied in practice for manykind of vibration problems

With respect to the motion path traditionally sloshingis analyzed for one-dimensional straight motion where thesloshing is often modeled as a simple pendulumThe motionplanning and sloshing suppression is relatively simple andstraightforward for this kind of problem [9 13] Researchesthat address suppression of sloshing or pendulum sway inhigher dimensional space are relatively few Tzamtzi et al[17] modeled the sloshing as simple pendulum in 2D planarvertical motionWilliams et al [18] used a dynamic program-ming approach to solve the sway-free motion of suspendedspherical pendulum in 2D planar horizontal motion Yanoand Terashima [14] proposedHSA approach to control slosh-ing in three degree of freedom Cartesian liquid containertransfer system

In this paper we propose the integration of the trajectoryplanning of seven degrees of freedom liquid container trans-fer robot arm and the sloshing suppression of the containedliquid We use the Mitsubishi PA 10-7C robot arm as theexperiment device The container is allowed to rotate aboutvertical axis while the pitch and roll angles are regulated tomaintain upright orientation of the liquid container Suchsystem setup is different from that used in other relatedpapers (eg [14 17 18]) in that translational motion in three-dimensional space plus orientation of the container needs tobe controlled The output of the proposed method is jointtrajectories to be used as a reference of the robot motionthus obviating the need to change the closed-loop controllerConsequently the proposed solution is easier to be applied todifferent kind of systems

Because the robot arm has spherical wrist the motioncan be conveniently decoupled into translational motion ofthe wrist locus and rotational motion of robot hand Thisdecomposition is useful in our application case becausefor sloshing suppression we want to take full control ofthe container orientation For the trajectory generation ofthe wrist translational motion we define the problem as anonlinear optimization model where each joint trajectoryis parameterized as cubic spline The locations of knots arepart of decision variables thus not fixed to allow obstacleavoidance in task space and quick motion It is an extensionto amore basic cubic spline optimization reported in our pre-vious paper [5] in two points motion in three-dimensionalspace is addressed by usingmultijoint robot arm and obstacleavoidance constraints are added As for the sloshing sup-pression we use a new strategy of applying input shapingThe decoupled translational and rotational motions are eachshaped by using its respective suitable input shaper andthen combined as the final trajectory We use the sphericalpendulummodel to simulate the response of the trajectory Inaddition a numerical sloshing simulation model is built forthe purpose of examining the sloshing behavior in responseto vertical motion of the liquid container in 3D space Finallyexperiments are carried out for a few example cases

2 Sloshing Model and Suppression

21 Equivalent Sloshing Model The theoretical natural fre-quency of sloshing inside an upright cylindrical container canbe derived from Navier-Stokes equations as [19]

1205962

119899= (

119892120585119899

119877

) tanh(120585119899ℎ

119877

) (1)

where 119892 is the gravity constant 119877 is the container radius ℎ isthe liquid height and 120585

119899is the root of the derivative of Bessel

function of the first kind where 119899 is the sloshing mode Forthe fundamental first mode 120585

1equals 1841

For control analysis the phenomenon of sloshing isoften modeled by its simpler mechanical equivalents forexample pendulum and mass-spring-damper system Herewe model the sloshing as a simple pendulum where thesway of the pendulum corresponds to the elevation of theliquid surface A simple pendulum system as depicted inFigure 1(a) consists of a point mass suspended by a masslessrigid cable of length 119871 to a friction-less movable support Forthe case where the support moves in straight line it can bedescribed by the following equation of motion

119871120579 + 119892 sin 120579 = minus cos 120579 (2)

where 119892 is the gravity constant and 119909 is the distance of thesupport from origin

In cases where the support moves in two-dimensionalhorizontal space the system can be modeled as a sphericalpendulum Figure 1(b) illustrates the spherical pendulummodel where the pendulum bob can freely swing in twodegrees of freedom spherical space as response to movementof the support in horizontal planar space Angle 120579 is the angle

Journal of Robotics 3

L

x

120579

(0 0)

(a)

X

Z

Y

120579

120579

120593

(b)

Figure 1 (a) Simple pendulum and (b) spherical pendulum

between plane ZY and the mass and 120593 is the angle in theperpendicular direction

The position of the pendulum bob in Cartesian coordi-nate system is

119909119901= minus119871 sin 120579 cos120593

119910119901= 119871 sin 120579

119911119901= minus119871 cos 120579 cos120593

(3)

The equations of motion of the spherical pendulum canbe derived as follows [18]

120579 = 2

120579 tan120593 minus

119892

119871

sin 120579cos120593

+

1

119871

cos 120579cos120593

= minus120579

2

tan120593 minus119892

119871

cos 120579 sin120593

minus

1

119871

( sin 120579 sin120593 + 119910 cos120593)

(4)

where 119909 and 119910 are the position of the support Indeed whenthere is no acceleration in Y direction and initially 120593 =

0 the above equations of motion will reduce to the simplependulum case in (2)

22 Input Shaping In the simple pendulum case the naturalfrequency and damping ratio can be obtained by linearizationof the equation of motion around equilibrium as

1205960= radic

119892

119871

120577 =

119888

2119898

radic

119871

119892

(5)

When excited by an impulse the sway response of thependulum is decaying sinusoidal

120579 (119905) = 119860

1205960

radic1 minus 1205772119890minus1205771205960119905 sin(120596

0119905radic1 minus 120577

2) (6)

Such sway can be suppressed by shaping the impulse inputsuch that the sum of the response is zero It is performedby splitting the original impulse into two or more impulsesand delays them such that their responses cancel each other[15 16] For example in the simplest case of splitting into twoimpulses the amplitudes of the first and second impulses are1198701 + 119870 and 11 + 119870 respectively where

119870 = 119890minus120577120587radic1minus120577

2

(7)

and time between impulse equals one-half damped period ofvibration

Δ119879 =

120587

1205960radic1 minus 120577

2 (8)

This set of impulses is called an input shaper and the two-impulse input shaper above is known as zero vibration (ZV)shaper A more robust input shaper type can be obtained byconvolving two ZV input shapers together resulting in three-impulse input shaper known as zero vibration and derivative(ZVD) input shaper Other than those basic input shapersthere are still many more input shapers developed withdifferent characteristics and capabilities to tackle differenttype of vibration problems

In a linear system any signal can be represented as acollection of scaled and shifted impulses Therefore the sys-tem response to a signal is the linear sum of the responsesof each component impulse It follows that the input shapingcan be applied by convolving the system input signal withthe desired input shaper as shown in Figure 2 As a resultthe motion time becomes longer as much as the length ofthe input shaper For example applying ZV input shaper


Signal Input shaper Shaped signal

Original signalShaped signal

lowast =

Figure 2 Shaping arbitrary signal by convolution

Table 1 Limit values of joint velocities and angles

Limit JointS1 S2 S3 E1 E2 W1 W2

119881MAX [rads] 1 1 2 2 2120587 2120587 2120587

119902UB [degree] 177 91 174 137 255 165 360

increases themotion time by half damped period of vibrationwhile with ZVD input shaper the motion time becomes onedamped period longer For general planar horizontal motionthe input signal may be in the form of acceleration or velocityreference of the motion

For systems whose motion is commanded by rotationalcommands for example rotary crane and other rotatingmechanical systems regular input shapers do not work wellfor suppressing motion induced vibration Those problemsexhibit different characteristics in that the angular commandimpulse excites sway in radial as well as tangential directionsAs the angular velocity increases the natural frequency bifur-cates from the nominal natural frequency resulting in twoswaymodes in the system For this kind of problem Lawrenceand Singhose [20] developed ZV2lin shaperThe input shaperconsists of three impulses where the amplitude and timingare determined by solving an optimization problem whichseeks minimization of vibration at the end of impulse trainThose are determined not only by natural frequency anddamping ratio but also by the nominal angular velocity androtation radius The input shaper length is approximatelyequal to one damped period more or less comparable to thelength of ZVD shaper

3 Trajectory Planning

The liquid transfer system concerned uses the Mitsubishi PA10-7C robot arm The frames placement and link dimensionof the robot arm are shown in Figure 3 Table 1 lists the limitvalue of each joint angle and angular velocity A cylindricalliquid container is attached to an extension at the robot tipThe diameter and height of the liquid container are 015mmand 025m respectively The distance between robot tip andthe center of the liquid container is 012 meter thus thelength of the last link is practically 02 meter For our currentapplication one of the arm joints the S3 joint is locked sothat only six joints are actively used Table 2 lists the Denavit-Hartenberg parameters of the robot arm configuration The

Table 2 Denavit-Hartenberg parameters

Parameter JointS1 S2 S3 E1 E2 W1 W2

120572 [rad] 1205872 minus1205872 1205872 minus1205872 1205872 minus1205872 0119889 [m] 0315 0 045 0 05 0 02119886 [m] 0 0 0 0 0 0 0

E1

W1

S1

E2

W2

S2

S3

X5

Y5

X3

Y3

X1

Y1

X6

Z6

X4

Z4

X2

Z2

X0

Z0

(a) Frames place-ment

031

5

045

05

015

02

025

X

Z

(b) Dimension of links and liquid container (in meter)

Figure 3 (a) Frames placement and (b) dimension of the liquid con-tainer system and the base frame

transformation matrix of a frame relative to the precedingframe is as follows

119860119894

= [

119894minus1119877119894

119894minus1119901119894

0 1

]

=

[[[[[

[

cos 120579119894minus sin 120579

119894cos120572119894

sin 120579119894sin120572119894119886119894cos 120579119894

sin 120579119894

cos 120579119894cos120572119894minus cos 120579

119894sin120572119894119886119894sin 120579119894

0 sin120572119894

cos120572119894

119889119894

0 0 0 1

]]]]]

]

(9)


The position and orientation of a link with respect toother link can be obtained by chain product of the transfor-mation matrices

119894119879119895= 119860119894+1119860119894+2sdot sdot sdot 119860119895= [

119894119877119895

119894119901119895

0 1

] (10)

The upper left 3 times 3 rotation matrix 119894119877119895describes the link

orientation while the vector 119894119901119895represents the position

The links E2 W1 and W2 form a spherical wrist wherethe three rotational joint axes intersect at a common pointThis robot configuration simplifies the inverse kinematiccalculation because it can be decoupled into two simplersubsystems The first three rotational links S1 S2 and E1provide the positioning in three-dimensional space while thespherical wrist adjusts the orientation of end effector

In the inverse kinematics for position the position ofwrist locus in the task space (ie 119909w 119910w and 119911w) is knownand the associated configuration of S1 S2 and E1 joints (ie1205791 1205792 and 120579

4) are to be sought The values can be calculated

geometrically as follows

1205791= atan2 (119910w 119909w)

1205794= atan2 (radic1 minus 1198632 119863)

1205792= atan2 (119911w minus 1198711 radic119909w2 + 119910w2)

minus atan2 (1198713sin 1205794 1198712+ 1198713cos 1205794) minus

120583

2

(11)

where 119871119894is the length of link 119894 atan2 is the arc tangent

function with two arguments and

119863 =

119909w2+ 119910w2+ (119911w minus 1198711)

2

minus 1198712

2minus 1198713

2

211987121198713

(12)

Meanwhile the inverse kinematics for orientation calcu-lates the configuration of E2 W1 and W2 joints (ie 120579

5 1205796

and 1205797) when the desired orientation of end effector relative

to the base frame (01198777) is known First from the inverse

kinematics for position the rotation matrix describing theorientation of the third link can be obtained (0119877

4) Then the

rotation operation of the wrist joints required to realize theend effector posture is calculated as

41198777= (01198774)

119879

(01198777) (13)

The three Euler angles can be calculated from the matrixand finally we can obtain the configuration of the wrist joints

1205795= atan2 (4119877

7 [2 3]

41198777 [1 3]) plusmn 120587 (14)

1205796= atan2 (radic1 minus 4119877

7[3 3]241198777 [3 3]) (15)

1205797= atan2 (4119877

7 [3 2]

41198777 [3 1]) plusmn 120587 (16)

where 41198777[119903 119888] is the element of the matrix 4119877

7at row 119903 and

column 119888When there aremore than one possibilities of anglevalues choose the one suitable for the application

Accordingly the trajectory planning of the liquid con-tainer transfer system is decoupled into planning of the trans-lation of the robot wrist position and rotation of the robothand The translation part is designed as joint space cubicspline trajectory while the hand rotation is designed in taskspace to maintain upright orientation of the liquid containerUpright orientation means that the container has only onedegree of freedom the rotation along the vertical z-axis offixed coordinate system Those translation and rotation tra-jectories are then combined tomake the final quick slosh-freetrajectory solution The trajectory planning steps are shownin Figure 4 which clearly depicts whether each step is done injoint space or task space Transformation between joint andtask space is carried out using direct and inverse kinematics

Figure 4(a) shows the overview of planning the transla-tional motion It starts with specifying the start point andend point as well as several knots between them in taskspace which are then transformed to joint space as the initialsolution to the cubic spline optimization The optimizationstep encompasses joint space as well as task space Althoughthe trajectory is generated in joint space it is possible to definetask space constraints in the optimization for example obsta-cle avoidance The obtained solution is joint space trajectorywhich is transformed back to task space where the commandshaping takes place The solution of the translational motionis obtained as joint trajectories of links S1 S2 and E1

Figure 4(b) shows the steps of planning the rotationalmotion of the robot handThe input is the start and end angleof the container along z-axis relative toXZ vertical planeTheshaped rotation trajectory is then transformed to joint spaceas the solution of the rotation part of the motion in form ofjoint trajectories of links E2 W1 and W2 Although rotationtrajectory generation is independent of translation trajectorygeneration it has to be performed after because the formerneeds the orientation history of the translational motion aswell as the totalmotion time in order to coordinate the overallmotion

31 TranslationalMotion Planning In this section we outlinethe cubic spline optimization approach for trajectory plan-ning It builds from the same basic approach as in a previouspaper [5] for two-dimensional problem and then adapted tothe current application of motion in three-dimensional spaceand is complemented with input shaping in the task space

A cubic spline trajectory of a robot joint is comprised ofseveral curve segments as illustrated in Figure 5 where eachsegment is represented by a cubic polynomial function oftimeThe general form of the cubic function in a segment 119894 is

119902119894(119905) = 119886

119894(119905 minus 119905119894)3

+ 119887119894(119905 minus 119905119894)2

+ 119888119894(119905 minus 119905119894) + 119889119894 (17)

where 119905 denotes the time which ranges from 119905119894to 119905119894+1

The curve segments are connected by knots Thus for a

trajectory that consists of 119899 curve segments there are 119899 + 1prespecified joint angle values 120579

119894 (119894 = 1 sdot sdot sdot 119899 + 1) which

consist of a start point an end point and 119899minus1 knots between


Joint space Task space

Start and end points knots

Initial solution

Cubic spline optimization

Trajectory in joint space

Trajectory of liquid container

Shaped trajectory of liquid container

Shaped trajectory of S1 S2 and E1 joints

Input shaping

(a) Trajectory planning for translation motion

Start and end orientation

Angular input shaping

Shaped trajectory of orientation

Shaped trajectory


of E2 W1 and W2 joints

(b) Trajectory planning for hand orientation

Figure 4 Trajectory planning steps for (a) translational motion and (b) hand orientation

Time

Join

t ang

le

1205791120579n+1

q1(t) qn(t)

t1 t2 tn tn+1

h1 hn

middot middot middot middot middot middot middot middot middot



Figure 5 Cubic spline trajectory

them The trajectory is required to pass through all thosepoints Other requirements are continuity in velocity andacceleration at every knot The relationship between jointangle 120579

119894 joint velocity

120579119894 joint acceleration

120579119894 and time

interval ℎ119894= (119905119894+1minus 119905119894) is as follows

120579119894ℎ119894+ 2

120579119894+1(ℎ119894+ ℎ119894+1) +

120579119894+2ℎ119894+1

= 6(

120579119894+2minus 120579119894+1

ℎ119894+1

minus

120579119894+1minus 120579119894

ℎ119894

)

(18)

In a joint trajectory with 119899 segments (or 119899 + 1 points)there are 119899 minus 1 such equations Together with the boundaryrequirements of acceleration ( 120579

1=120579119899+1= 0) we have a total

of 119899+1 equationsThe robotmotion usually also requires thatvelocities at start and end points equal zero To accommodatethis two more unknowns are added by inserting two virtualpoints one after the start point and another one just before

the end point Furthermore because 1205791and

120579119899+1

are alreadyknown as zero they can be removed from the equations Jointangles 120579

119894are defined beforehand by the trajectory designer

The knots may be specified as points in Cartesian task spaceand later transformed into joint space by inverse kinematicsThe above system of linear equations forms a symmetrictridiagonal system which can be solved efficiently using thetridiagonal matrix algorithm (TDMA) to find acceleration ateach knot

The trajectories of S1 S2 and E1 joints are each designedas piecewise cubic splinesThe joint trajectories are coupled toeach other by using the same value of time interval variablesℎ119894 We formulate the trajectory generation problem as a

nonlinear optimization problem The objective function iscombination of motion time and weighted squared accelera-tionThere are several constraints to consider in the trajectorygeneration robot position in the task space joint velocity andjoint jerk According to (18) there are three kinds of variablesinvolved segment time joint acceleration and joint angleat knots We let both the segment time and the joint angles(knots location) as the optimization variables The optimi-zation problem is thus as follows

minx 119865 =

119899

sum

119894=1

ℎ119894+ 119908119886int 1199022

119894119889119905

subject to 10038161003816100381610038161003816119902lowast

119894119895

10038161003816100381610038161003816le 119902

MAX119895

119894 = 1 sdot sdot sdot 119899 119895 = 1 sdot sdot sdot 119898

10038161003816100381610038161003816Vlowast119894119895

10038161003816100381610038161003816le VMAX119895



X

Y

(a) Perspective view

X

Y

(b) Top view (X-Y plane)

Figure 6 Motion path of example case of translation motion

10038161003816100381610038161003816

120579

10038161003816100381610038161003816le 119869

MAX

119894 = 1 sdot sdot sdot 119899 + 1 119895 = 1 sdot sdot sdot 119898

119889119894(119871119895 119874) ge 119903

119871119895+ 119903119874


(19)

where x is the vector of optimization variables which containsthe segment times as well as the joint angles excluding thefixed start point end point and the two virtual points

x = (ℎ1 ℎ

119899 120579119894119895 120579

119899119898)

119894 = 3 sdot sdot sdot 119899 minus 1 119895 = 1 sdot sdot sdot 119898

(20)

Here all internal knots are part of the optimization vari-ables instead of predefined pointsThis provides flexibility forthe optimization to generate quick motion as well as obstacleavoidance at the same time If the plannedmotion is requiredto pass certain fixed points on the way they can be added asadditional equality constraintsThe first term of the objectivefunction is to minimize the total motion time which is themain objectiveThe second term is used to avoid unnecessarymotion of robot joints where 119908

119886is the weight parameter In

our subsequent simulations and experiments the parametervalue equals 1119890minus4

The first and second constraints are joint angle andvelocity constraints according to the physical joint limits inTable 1The third is jerk constraint to prevent excessive valueof joint jerk The maximum jerk of each joint is set equal to15ms3 The last constraint is for obstacle avoidance wherethe obstacle and robot links are modeled as rigid capsuleswith radius 119903

119874and 119903119871119895 respectively The function 119889

119894(119871119895 119874)

is the minimum distance between link 119871119895and static obstacle

119874 in segment 119894 The obstacle and linksrsquo axis are discretizedinto several representative pointsThe distance between a linkand obstacle is defined as the minimum distance between thepoints in the obstacle and the points in the link

Weuse theMATLABOptimizationToolbox implementa-tion of Sequential Quadratic Programming (SQP) for solving

the above constrained nonlinear trajectory planning opti-mization problemThe SQP is one of the most used methodsin solving general constrained nonlinear optimization prob-lems The method solves the problem by iteratively solving aquadratic subproblem which is the quadratic approximationof the Lagrange function of the original nonlinear problemThe solution of the quadratic subproblem is then used to forma new set of optimization variables for the next iteration of theoriginal problem The calculation time differs depending onthe computer used for calculation it is around 15 seconds fora typical problem size in our Core2 Duo computer

Figure 6 shows an example motion path of a cubictrajectory generated by the abovemethod in perspective viewand top viewThe PA-10 robot arm is stationed at origin of theworld frameThe robot has to move its wrist from start point[03 minus08 02] to end point [05 05 02] in the task spacewhich correspond to 120579START = [minus121 minus125 minus087] rad and120579END = [079 minus097 minus144] rad in the joint space The robotposture shown in the figure is the configuration at start pointA cylindrical obstacle with radius 005m lies vertically inthe workspace at [05 minus03] For this problem four initialvia points are specified randomly between the start and endpoints The line going from the start point and the endpoint is the generated motion path in the task space whichsuccessfully avoids the obstacleThe small spheres on the linedenote the location of output via points The total motiontime is 27 seconds Figure 7 shows the joint trajectories ofthe example case the piecewise cubic joint angles and thepiecewise quadratic joint velocities The small circles in thefigure denote the respective values at the knots includingat the two virtual knots From the velocity figure we canunderstand that the motion time is limited by joint S1 wherethe velocity hits its limit most of the time

The next step in the translational motion planning is tosuppress the sloshing generated by the trajectory obtainedabove For analysis we use the pendulummodel in Section 21to represent the sloshing where the pendulum support isattached at the robot wrist locus and the pendulum swaysfreely as the robot moves For the case where the pendulumsupport can move in a horizontal planar space we firstdecompose the input signal to its projection into orthogonal


02

Joint S1

01

Joint S2

0

2Joint E1

Time (s)

Join

t ang

le

minus2

minus2

minus1

0 05 1 15 2 25

0 05 1 15 2 25

0 05 1 15 2 25

(rad

)Jo

int a

ngle

(rad

)Jo

int a

ngle

(rad

)

(a) Joint angle

0

1

0

1

0

2

Join

t vel

ocity

minus2

minus1

minus1

Joint S1

Joint S2

Joint E1

Time (s)

0 05 1 15 2 25

0 05 1 15 2 25

0 05 1 15 2 25

(rad

s)

Join

t vel

ocity

(rad

s)

Join

t vel

ocity

(rad

s)

(b) Joint velocity

Figure 7 Trajectory of example case of translational motion (a) joint angle (b) joint velocity

Cartesian axes and then apply input shaping independentlyto each input signal By doing this the spherical pendulumis considered as two independent simple pendulums whichact on orthogonal vertical planes such that the sway behaviorin one subsystem is not affected by the states in the othersubsystem This simplified approach works well in smallangle regions and indeed linearization of (4) by small angleapproximations and removing quadratic and higher termsresults in two independent equations of simple pendulum

We use the ZVD input shaping for sloshing suppressionFor that the velocity trajectory is decomposed into two or-thogonal velocities parallel to x-axis and y-axis respectivelyEach velocity trajectory is then convolved individually withZVD input shaper The shaped results are then added backand integrated to produce the shaped trajectory in task spaceJoint space solution can be obtained by inverse kinematics

As an example we simulate a spherical pendulum oflength 02 meter moving according to the trajectory obtainedabove The natural frequency and the vibration period ofthe pendulum model are 70 radsec and 08973 secondsrespectively In sloshing problems the damping is usuallyvery small therefore here we assume zero damping in thependulum model Figure 8 shows the original and ZVDshapedCartesian velocities inX andY directionsThemotiontime is 27 seconds and 36 seconds in the unshaped and ZVDshaped case respectively Figure 9 shows the sway angles inthe spherical coordinate system At the end ofmotion sway ismuch suppressed in the shaped trajectory Figure 10 comparesthe projection of the pendulum position at horizontal planefor the two cases original and shaped trajectories It isclear from the figure that the shaped trajectory generatessignificantly less sway than its unshaped counterpart

32 Rotational Motion Planning The liquid container has tobe maintained upright all time This means that there is only

one task space DOF the rotation along the vertical z-axisThus the hand rotationalmotion can be simply parameterizedby the angle along z-axis relative to the XZ plane Here robothand means any parts after the wrist including the last robotlink the container holder and the liquid container itself Thelength of the robot hand is the distance between the wristlocus and the center of container which equals 02 meterThetrajectory is generated by applying ZV2lin input shaping to arectangular velocity profile in which the maximum angularvelocity is

119904 =

120574119891minus 120574119904

119879

(21)

where 119879 is the time of the unshaped translation motion and120574119904and 120574119891are the desired start and finish angle value respec-

tively This is to coordinate the translational and rotationalmotions so that both motions end at approximately the sametime

As an example we use the same spherical pendulummodel to simulate the motion of the pendulum support ona circular trajectory of length 1205872 radian where the robothand moves from angle minus1205872 radian to 0 (relative to XZplane) The unshaped translation motion time is 27 secondsthus the maximum angular velocity for the rotational motionequals 058 radsec Upon applying ZV2lin input shaper therotational motion time becomes 347 seconds

Figure 11 shows the comparison of the projections of thependulum bob on XY plane of the unshaped trajectory andthe trajectory shaped by ZV2lin input shaper The circularmotion starts from the origin and stops at (02 02) In theshaped case the pendulum sway ismuch suppressed both thesway while the support is moving and the residual sway afterthe support stops This shows the effectiveness of the shaperin suppressing vibration for motions generated by angularcommand


0 05 1 15 2 25 3 35

0

01

02

03

04

05

06

Time (s)

Velo

city

(ms

)

OriginalZVD shaped

minus01

minus02

minus03

minus04

(a) X direction

Velo

city

(ms

)

0

02

04

06

08

1

minus020 05 1 15 2 25 3 35

Time (s)

OriginalZVD shaped

(b) Y direction

Figure 8 Original and ZVD shaped velocity in task space (a) X direction (b) Y direction

0 05 1 15 2 25 3 35 4 45 5

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

120579

120601

Time (s)

(a) Without shaping

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)

(b) ZVD input shaping

Figure 9 Pendulum sway angles (a) without shaping (b) with ZVD shaping

The trajectories of the three spherical wrist links can beobtained using the inverse kinematic relations in (13)ndash(16)the shaped angle value and the orientation history of thetranslationmotion partThe rotationmatrix 0119877

7is the desired

orientation of the liquid container

01198777=[[

[

cos 120574 minus sin 120574 0sin 120574 cos 120574 0

0 0 1

]]

]

(22)

where 120574 is the angle about z-axis

The translational and rotational motions are independentas each is handled by different set of robot joints Because ofthat the final trajectory can be obtained simply by combiningthe joint trajectories of the translational and rotationalmotions Once again we simulate cases where sphericalpendulum is attached to the robot tip where the robot wristlocus moves from location (03 minus08 02) to (05 05 02)while at the same time the robot hand rotates with respectto the wrist from angle minus1205872 rad to 0 rad In short thetrajectory is the combination of the simulated trajectories atSections 31 and 32 Figure 12 compares the sway angles ofthe case without shaping and the shaped case The shaping


X

Y

08 06 04 02 00

02

04

06

08

1

Robot wrist positionProjection to XY plane

minus02 minus04 minus06 minus08 minus1

(a) Without shaping

X

Y

08 06 04 02 00

02

04

06

08

1




Figure 10 Projection of pendulum bob position in XY plane (a) without shaping (b) with ZVD shaping

0 005

0

005

01

015

02

025

03

X

Y

minus025

minus005

minus01minus02 minus015 minus01 minus005

Support pointProjection to XY plane

(a) Without shaping


0 005

0

005

01

015

02

025

03

X

Y

minus025

minus005


(b) ZV2lin shaped

Figure 11 Pendulum position in XY plane of circular motion for (a) without shaping (b) with shaping

is done individually the translation motion is shaped usingZVD input shaper and the rotational motion is shaped usingZV2lin input shaping The motion time is the same as inSection 31 27 seconds and 36 seconds for the unshaped andshaped case respectively Figure 13 shows the motion path ofthe robot tip and the projection of the pendulum bob in XYplane From those two figures we can understand that thestrategy of combining individually shaped trajectories resultin much reduced sway of the pendulum In that way fordifferent conditions (in this case translational and rotationalmotion) we can use suitable input shaping for each conditionand then combined the generated trajectories

4 Numerical Simulation of Sloshing

In order to analyze the effect of vertical motion to sloshingwe built a numerical simulation of sloshing inside a mov-ing liquid container based on exact distributed parametercomputational fluid dynamics (CFD) model not the simplependulum model The liquid container is cylindrical with150mm diameter and 250mm in height The depth of liquidis 170mm The simulation model is developed in the opensource OpenFOAM software package The liquid oscillationis measured during simulation time with measurement sam-pling time 001 seconds by two virtual probes that record the


Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)

(a) Without shaping

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)

(b) Combined input shaping

Figure 12 Pendulum sway angles (a) without shaping (b) combined input shaping

Center of container

X

Y

08 06 04 02 00

02

04

06

08

1

Projection to XY plane


(a) Without shaping

X

0

02

04

06

08

1

Projection to XY planeCenter of container

Y

08 06 04 02 0 minus02 minus04 minus06 minus08 minus1


Figure 13 Projection of pendulum bob position in XY plane (a) without shaping (b) combined input shaping

liquid height near the container wall at the direction of x-axisand y-axis respectively By using the measured sloshing datawe identified the dominant fundamentalmode of the sloshingat frequency 23Hz

As explained in Section 31 input shaping for translationalmotion is implemented by decomposing the trajectory intotwo Cartesian components (x-axis and y-axis) and thenapplying input shaping to those two trajectory componentsindividually The shaped trajectories are then combined asthe final shaped trajectory By using the spherical pendulummodel it has been shown that the approach works well insuppressing the pendulum sway However when we considermotion in 3D space the motion component in z-axis alsohas to be taken into consideration To examine the effect ofacceleration in z-axis toward sloshing we run a simulationcomparing trajectorieswith andwithout vertical acceleration

In both trajectories the liquid container moves horizontallyfor the first one second to generate some sloshing Thenin trajectory A we let the container stay still while in tra-jectory B the liquid container moves upward Figure 14shows the comparison of the sloshing generated by bothtrajectories From second one the sloshing frequency differsslightly where it is a bit higher when the liquid containeraccelerates upward Figure 15 shows the comparison infrequency domain This simulation result agrees with thetheory that states natural frequency of sloshing is directlyproportional to square root of vertical acceleration (1) Inthe usual case of lateral movement it consists of only theconstant gravitational acceleration But when it acceleratesupward the vertical acceleration would be higher than 119892 andas a result natural frequency becomes higher The theoreticalnatural frequency under constant vertical acceleration of


0 1 2 3 4 5

0

0005

001

0015

Time (s)

Liqu

id su

rface

disp

lace

men

t (m

)

Without verticalWith vertical

minus0005

minus001

Figure 14 Comparison of the simulated sloshing under verticalacceleration

0 2 4 6 8 100

05

1

15

2

Frequency (Hz)

FFT

ampl

itude

23926

22949


Figure 15 Comparison of the sloshing frequency under verticalacceleration

1ms2 equals 24Hz which is approximately equal with thesimulation result

The consequence of this to the sloshing suppression isthat we have to consider the frequency shift in the inputshaper calculation From simulation we understand that therange of the frequency shift is not wide With currentsetup of the robot arm the vertical acceleration of a typicalmotion is under plusmn4ms2 which approximately correspondsto maximum frequency shift of plusmn04Hz To deal with thissmall shift we use zero vibration and damping (ZVD) inputshaper which is a more robust version of the input shaperinstead of the basic ZV shaper

0 1 2 3 4 5

0

002

004

006

Liqu

id su

rface

disp

lace

men

t (m

)

OriginalZV shaped

ZVD shaped

Time (s)

minus002

minus004

minus006

Figure 16 Comparison of sloshing in vertically dominant case

Y

(08 01 03)

(01 09 02)

(05 05 01)

X (03 minus08 02)

Figure 17 Motion path of the experiment cases

To examine the sloshing in a vertical dominant motiontrajectory we setup another simulation case where liquidcontainer is moved from location (02 06 02) to loca-tion (03 02 11) In this trajectory vertical movement isrelatively dominant Thus we expect the natural frequencyto shift along the trajectory Figure 16 shows the sloshingcomparison Consistent with the previous example casethe shaped trajectories result in less vibration In additionthe robust ZVD shaped trajectory improves the sloshingsuppression The maximum residual sloshing is 4622mm404mm and 176mm for original ZV shaped and ZVDshaped trajectories respectively From this simulation studywe understand that vertical motion affects the sloshing inthat small shift of natural frequency occurs The small shiftjustifies the use of the zero vibration and derivative (ZVD)input shaping which ismore robust than the default ZV inputshaping in our proposed solution


Disp

lace

men

t (m

m)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD

End of ldquocubic + ZVDrdquo motion

End of ldquocubicrdquo motion

(a) Case 1D

ispla

cem

ent (

mm

)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(b) Case 2

100

120

140

160

180

200

220

240

Disp

lace

men

t (m

m)



0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD

(c) Case 3

Figure 18 Sloshing measurement of (a) Case 1 (b) Case 2 and (c) Case 3

5 Experiment Results and Discussion

In order to assess the effectiveness of the overall proposedsystem including the trajectory planning for translationaland rotational motion as well as the sloshing we setup a fewexperiment cases Table 3 lists the start and end position andorientation The start points and end points are the positionof the center of the container liquidThe angles are measuredrelative to XZ plane Figure 17 illustrates the motion path ofthe three experiment cases where the start and end points ofeach case are shown by small spheres

The generated sloshing as measured by level sensor isshown in Figure 18 It is measured for ten seconds from thestart of motion The motion time residual vibration and

Table 3 The start and end configuration of the experiment cases

Case Point AngleStart End Start End

1 (03 minus08 02) (08 01 03) minus1205872 02 (08 01 03) (01 09 02) 0 12058723 (01 09 02) (05 06 09) 1205872 0

vibration reduction of the three cases are shown in Table 4The motion time is the total time required to move fromthe start point to end point The maximum residual sloshingfor both unshaped and ZVD shaped trajectories is measured


Table 4 Result of experiments

Case Motion time [s] Max residual sloshing [mm] Sloshing reduction []Unshaped ZVD Unshaped ZVD

1 1791 2197 25735 6123 762072 1783 2189 221 5929 731723 1156 1562 45997 9921 78431

from the ZVD shaped motion time until the end of measure-ment so as to make fair comparison

In general we can see that the shaped trajectories generatemuch less sloshing at expense of longer motion times Theadditional motion time is 0406 seconds for all cases Thatamount is equivalent to the length of the ZVD input shaperwhich equals one vibration period The sloshing reduction ismore or less the same in all cases where the largest is as muchas 78431 in Case 3 The motion path of Case 3 is mostly inupward direction This shows that the solution does not haveproblem in handling trajectory that contains vertical motion

6 Conclusion and Further Work

This paper has examined the case of liquid container transfersystem using multi joint robot arm where quick motionand minimal sloshing are concurrently addressed The pro-posed framework integrates trajectory planning in three-dimensional space and input shaping for sloshing suppres-sion The trajectory planning algorithm based on cubicspline optimization has been built to generate trajectoriesfor translation motion in three-dimensional space with jointkinematic constraints and obstacle avoidance The sloshingsuppression is built into the algorithm based on principle ofinput shaping and decoupling of the robot motion into trans-lation and hand rotation subsystems It has been shown usingsimulations and experiments that the generated trajectoriesinduce much less sloshing thus demonstrating the benefit ofthe suppression strategy Furtherworkwillmainly be directedto improve the current framework by incorporating thedynamics consideration into the algorithm Another plannedimprovement is to provide redundancy by utilizing oneadditional joint of the robot arm for better maneuverability

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C-S Lin P-R Chang and J Y S Luh ldquoFormulation and opti-mization of cubic polynomial joint trajectories for industrialrobotsrdquo IEEE Transactions on Automatic Control vol 28 no 12pp 1066ndash1074 1983

[2] A Piazzi and A Visioli ldquoGlobal minimum-jerk trajectory plan-ning of robot manipulatorsrdquo IEEE Transactions on IndustrialElectronics vol 47 no 1 pp 140ndash149 2000

[3] T Chettibi H E Lehtihet M Haddad and S Hanchi ldquoMini-mum cost trajectory planning for industrial robotsrdquo EuropeanJournal ofMechanicsmdashASolids vol 23 no 4 pp 703ndash715 2004

[4] A Gasparetto and V Zanotto ldquoA technique for time-jerkoptimal planning of robot trajectoriesrdquo Robotics and Computer-Integrated Manufacturing vol 24 no 3 pp 415ndash426 2008

[5] W Aribowo and K Terashima ldquoCubic spline trajectory plan-ning and vibration suppression of semiconductor wafer transferrobot armrdquo International Journal of Automation Technology vol8 no 2 pp 265ndash274 2014

[6] K Yano and K Terashima ldquoRobust liquid container transfercontrol for complete sloshing suppressionrdquo IEEE Transactionson Control Systems Technology vol 9 no 3 pp 483ndash493 2001

[7] M Reyhanoglu and J Rubio Hervas ldquoNonlinear modelingand control of slosh in liquid container transfer via a PPRrobotrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 6 pp 1481ndash1490 2013

[8] J Feddema C Dohrmann and G Parker ldquoA comparison ofmaneuver optimization and input shaping filters for roboticallycontrolled slosh-free motion of an open container of liquidrdquo inProceedings of the American Control Conference AlbuquerqueNM USA 1997

[9] L Consolini A Costalunga A Piazzi and M Vezzosi ldquoMin-imum-time feedforward control of an open liquid containerrdquo inProceedings of the 39th Annual Conference of the IEEE IndustrialElectronics Society (IECON rsquo13) pp 3592ndash3597 IEEE ViennaAustria November 2013

[10] K Terashima and K Yano ldquoSloshing analysis and suppressioncontrol of tilting-type automatic pouring machinerdquo ControlEngineering Practice vol 9 no 6 pp 607ndash620 2001

[11] M Hamaguchi and T Taniguchi ldquoTransfer control and curvedpath design for cylindrical liquid containerrdquo in Proceedings of15th IFAC World Congress pp 1479ndash1479 Barcelona Spain2002

[12] A Aboel-Hassan M Arafa and A Nassef ldquoDesign and opti-mization of input shapers for liquid slosh suppressionrdquo Journalof Sound and Vibration vol 320 no 1-2 pp 1ndash15 2009

[13] W Aribowo T Yamashita K Terashima and H KitagawaldquoInput shaping control to suppress sloshing on liquid containertransfer using multi-joint robot armrdquo in Proceedings of the 23rdIEEERSJ 2010 International Conference on Intelligent Robotsand Systems (IROS rsquo10) pp 3489ndash3494 Taipei Taiwan October2010

[14] K Yano and K Terashima ldquoSloshing suppression controlof liquid transfer systems considering a 3-D transfer pathrdquoIEEEASME Transactions on Mechatronics vol 10 no 1 pp 8ndash16 2005

[15] N C Singer andW P Seering ldquoPreshaping command inputs toreduce system vibrationrdquo Journal of Dynamic Systems Measure-ment and Control vol 112 no 1 pp 76ndash82 1990


[16] T Singh and S R Vadali ldquoRobust time-delay controlrdquo Journalof Dynamic Systems Measurement and Control vol 115 no 2App 303ndash306 1993

[17] M P Tzamtzi F N Koumboulis and N D Kouvakas ldquoA twostage robot control for liquid transferrdquo in Proceedings of the 12thIEEE International Conference on Emerging Technologies andFactory Automation pp 1324ndash1333 Patras Greece September2007

[18] C Williams G Starr J Wood and R Lumia ldquoCurvilineartransport of suspended payloadsrdquo in Proceedings of the IEEEInternational Conference on Robotics and Automation (ICRArsquo07) pp 4537ndash4543 IEEE Roma Italy April 2007

[19] R Ibrahim Liquid Sloshing Dynamics Theory and ApplicationsCambridge University Press New York NY USA 2005

[20] J Lawrence and W Singhose ldquoCommand shaping slewingmotions for tower cranesrdquo Journal of Vibration and Acousticsvol 132 no 1 Article ID 011002 11 pages 2010

International Journal of

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RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



knots is not fixed which is suitable for trajectory planningproblem with obstacle avoidance In addition it often resultsin a quicker motion because there is less restriction for theoptimization algorithm to reach optimal velocity profile

The smooth cubic spline trajectories induce less vibrationwhen compared to nonsmooth or lower order trajectoriesHowever in cases where minimal sloshing is requiredsmooth trajectory alone is not enough and it needs to besupplemented with explicit sloshing control strategy Therehave been numerous researches dealing with sloshing sup-pression in liquid container transfer system Feedback controlbased approaches present good and relatively robust sloshingcontrol ability [6 7] However in some practical casessensor measurements cannot always be reliable or even aredifficult to perform for example in the case of controllingsloshing of high-temperature molten metal In these casesfeedforward approaches [8ndash14] are more useful Based onsloshing models the behavior of the system can be predictedand thus the trajectory or motion path can be designedaccordingly Examples include generalized bang-bang control[9] infinite impulse response (IIR) filter [8] input shapingfilter [10ndash13] andHybrid Shape Approach filters [14] Amongthem the input shaping approach is particularly interestingbecause of its simplicity and effectiveness in practice It worksby decomposing the command input signal into two ormoreusing only the knowledge of natural frequency and dampingratio of the system [15 16] Over time various type of inputshapers with different characteristics have been proposedand they have been successfully applied in practice for manykind of vibration problems

With respect to the motion path traditionally sloshingis analyzed for one-dimensional straight motion where thesloshing is often modeled as a simple pendulumThe motionplanning and sloshing suppression is relatively simple andstraightforward for this kind of problem [9 13] Researchesthat address suppression of sloshing or pendulum sway inhigher dimensional space are relatively few Tzamtzi et al[17] modeled the sloshing as simple pendulum in 2D planarvertical motionWilliams et al [18] used a dynamic program-ming approach to solve the sway-free motion of suspendedspherical pendulum in 2D planar horizontal motion Yanoand Terashima [14] proposedHSA approach to control slosh-ing in three degree of freedom Cartesian liquid containertransfer system

In this paper we propose the integration of the trajectoryplanning of seven degrees of freedom liquid container trans-fer robot arm and the sloshing suppression of the containedliquid We use the Mitsubishi PA 10-7C robot arm as theexperiment device The container is allowed to rotate aboutvertical axis while the pitch and roll angles are regulated tomaintain upright orientation of the liquid container Suchsystem setup is different from that used in other relatedpapers (eg [14 17 18]) in that translational motion in three-dimensional space plus orientation of the container needs tobe controlled The output of the proposed method is jointtrajectories to be used as a reference of the robot motionthus obviating the need to change the closed-loop controllerConsequently the proposed solution is easier to be applied todifferent kind of systems

Because the robot arm has spherical wrist the motioncan be conveniently decoupled into translational motion ofthe wrist locus and rotational motion of robot hand Thisdecomposition is useful in our application case becausefor sloshing suppression we want to take full control ofthe container orientation For the trajectory generation ofthe wrist translational motion we define the problem as anonlinear optimization model where each joint trajectoryis parameterized as cubic spline The locations of knots arepart of decision variables thus not fixed to allow obstacleavoidance in task space and quick motion It is an extensionto amore basic cubic spline optimization reported in our pre-vious paper [5] in two points motion in three-dimensionalspace is addressed by usingmultijoint robot arm and obstacleavoidance constraints are added As for the sloshing sup-pression we use a new strategy of applying input shapingThe decoupled translational and rotational motions are eachshaped by using its respective suitable input shaper andthen combined as the final trajectory We use the sphericalpendulummodel to simulate the response of the trajectory Inaddition a numerical sloshing simulation model is built forthe purpose of examining the sloshing behavior in responseto vertical motion of the liquid container in 3D space Finallyexperiments are carried out for a few example cases

2 Sloshing Model and Suppression

21 Equivalent Sloshing Model The theoretical natural fre-quency of sloshing inside an upright cylindrical container canbe derived from Navier-Stokes equations as [19]

1205962

119899= (

119892120585119899

119877

) tanh(120585119899ℎ

119877

) (1)

where 119892 is the gravity constant 119877 is the container radius ℎ isthe liquid height and 120585

119899is the root of the derivative of Bessel

function of the first kind where 119899 is the sloshing mode Forthe fundamental first mode 120585

1equals 1841

For control analysis the phenomenon of sloshing isoften modeled by its simpler mechanical equivalents forexample pendulum and mass-spring-damper system Herewe model the sloshing as a simple pendulum where thesway of the pendulum corresponds to the elevation of theliquid surface A simple pendulum system as depicted inFigure 1(a) consists of a point mass suspended by a masslessrigid cable of length 119871 to a friction-less movable support Forthe case where the support moves in straight line it can bedescribed by the following equation of motion

119871120579 + 119892 sin 120579 = minus cos 120579 (2)

where 119892 is the gravity constant and 119909 is the distance of thesupport from origin

In cases where the support moves in two-dimensionalhorizontal space the system can be modeled as a sphericalpendulum Figure 1(b) illustrates the spherical pendulummodel where the pendulum bob can freely swing in twodegrees of freedom spherical space as response to movementof the support in horizontal planar space Angle 120579 is the angle


L

x

120579

(0 0)

(a)

X

Z

Y

120579

120579

120593

(b)




119909119901= minus119871 sin 120579 cos120593

119910119901= 119871 sin 120579

119911119901= minus119871 cos 120579 cos120593

(3)


120579 = 2

120579 tan120593 minus

119892

119871

sin 120579cos120593

+

1

119871

cos 120579cos120593

= minus120579

2

tan120593 minus119892

119871

cos 120579 sin120593

minus

1

119871

( sin 120579 sin120593 + 119910 cos120593)

(4)




1205960= radic

119892

119871

120577 =

119888

2119898

radic

119871

119892

(5)


120579 (119905) = 119860

1205960


0119905radic1 minus 120577

2) (6)



2

(7)


Δ119879 =

120587

1205960radic1 minus 120577

2 (8)






lowast =




119881MAX [rads] 1 1 2 2 2120587 2120587 2120587

119902UB [degree] 177 91 174 137 255 165 360








E1

W1

S1

E2

W2

S2

S3

X5

Y5

X3

Y3

X1

Y1

X6

Z6

X4

Z4

X2

Z2

X0

Z0


031

5

045

05

015

02

025

X

Z




119860119894

= [

119894minus1119877119894

119894minus1119901119894

0 1

]

=

[[[[[

[

cos 120579119894minus sin 120579

119894cos120572119894

sin 120579119894sin120572119894119886119894cos 120579119894

sin 120579119894

cos 120579119894cos120572119894minus cos 120579

119894sin120572119894119886119894sin 120579119894

0 sin120572119894

cos120572119894

119889119894

0 0 0 1

]]]]]

]

(9)



119894119879119895= 119860119894+1119860119894+2sdot sdot sdot 119860119895= [

119894119877119895

119894119901119895

0 1

] (10)







1205791= atan2 (119910w 119909w)

1205794= atan2 (radic1 minus 1198632 119863)



120583

2

(11)



119863 =

119909w2+ 119910w2+ (119911w minus 1198711)

2

minus 1198712

2minus 1198713

2

211987121198713

(12)


5 1205796




4) Then the


41198777= (01198774)

119879

(01198777) (13)


1205795= atan2 (4119877

7 [2 3]

41198777 [1 3]) plusmn 120587 (14)

1205796= atan2 (radic1 minus 4119877

7[3 3]241198777 [3 3]) (15)

1205797= atan2 (4119877

7 [3 2]

41198777 [3 1]) plusmn 120587 (16)


7at row 119903 and







119902119894(119905) = 119886

119894(119905 minus 119905119894)3

+ 119887119894(119905 minus 119905119894)2

+ 119888119894(119905 minus 119905119894) + 119889119894 (17)









Initial solution






Input shaping





Shaped trajectory





Time

Join

t ang

le

1205791120579n+1

q1(t) qn(t)

t1 t2 tn tn+1

h1 hn








120579119894 and time


120579119894ℎ119894+ 2

120579119894+1(ℎ119894+ ℎ119894+1) +

120579119894+2ℎ119894+1

= 6(

120579119894+2minus 120579119894+1

ℎ119894+1

minus

120579119894+1minus 120579119894

ℎ119894

)

(18)


1=120579119899+1= 0) we have a total



120579119899+1






minx 119865 =

119899

sum

119894=1

ℎ119894+ 119908119886int 1199022

119894119889119905


119894119895

10038161003816100381610038161003816le 119902

MAX119895


10038161003816100381610038161003816Vlowast119894119895

10038161003816100381610038161003816le VMAX119895



X

Y


X

Y



10038161003816100381610038161003816

120579

10038161003816100381610038161003816le 119869

MAX


119889119894(119871119895 119874) ge 119903

119871119895+ 119903119874


(19)


x = (ℎ1 ℎ

119899 120579119894119895 120579

119899119898)


(20)






119894(119871119895 119874)








02

Joint S1

01

Joint S2

0

2Joint E1

Time (s)

Join

t ang

le

minus2

minus2

minus1

0 05 1 15 2 25

0 05 1 15 2 25

0 05 1 15 2 25

(rad

)Jo

int a

ngle

(rad

)Jo

int a

ngle

(rad

)

(a) Joint angle

0

1

0

1

0

2

Join

t vel

ocity

minus2

minus1

minus1

Joint S1

Joint S2

Joint E1

Time (s)

0 05 1 15 2 25

0 05 1 15 2 25

0 05 1 15 2 25

(rad

s)

Join

t vel

ocity

(rad

s)

Join

t vel

ocity

(rad

s)

(b) Joint velocity







119904 =

120574119891minus 120574119904

119879

(21)






0 05 1 15 2 25 3 35

0

01

02

03

04

05

06

Time (s)

Velo

city

(ms

)

OriginalZVD shaped

minus01

minus02

minus03

minus04

(a) X direction

Velo

city

(ms

)

0

02

04

06

08

1

minus020 05 1 15 2 25 3 35

Time (s)

OriginalZVD shaped

(b) Y direction


0 05 1 15 2 25 3 35 4 45 5

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

120579

120601

Time (s)

(a) Without shaping

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)




7is the desired


01198777=[[

[


0 0 1

]]

]

(22)




X

Y

08 06 04 02 00

02

04

06

08

1



(a) Without shaping

X

Y

08 06 04 02 00

02

04

06

08

1





0 005

0

005

01

015

02

025

03

X

Y

minus025

minus005



(a) Without shaping


0 005

0

005

01

015

02

025

03

X

Y

minus025

minus005


(b) ZV2lin shaped






Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)

(a) Without shaping

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)



Center of container

X

Y

08 06 04 02 00

02

04

06

08

1



(a) Without shaping

X

0

02

04

06

08

1


Y








0 1 2 3 4 5

0

0005

001

0015

Time (s)

Liqu

id su

rface

disp

lace

men

t (m

)


minus0005

minus001


0 2 4 6 8 100

05

1

15

2

Frequency (Hz)

FFT

ampl

itude

23926

22949





0 1 2 3 4 5

0

002

004

006

Liqu

id su

rface

disp

lace

men

t (m

)

OriginalZV shaped

ZVD shaped

Time (s)

minus002

minus004

minus006


Y

(08 01 03)

(01 09 02)

(05 05 01)

X (03 minus08 02)




Disp

lace

men

t (m

m)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(a) Case 1D

ispla

cem

ent (

mm

)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(b) Case 2

100

120

140

160

180

200

220

240

Disp

lace

men

t (m

m)



0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD

(c) Case 3







1 (03 minus08 02) (08 01 03) minus1205872 02 (08 01 03) (01 09 02) 0 12058723 (01 09 02) (05 06 09) 1205872 0





1 1791 2197 25735 6123 762072 1783 2189 221 5929 731723 1156 1562 45997 9921 78431







References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










L

x

120579

(0 0)

(a)

X

Z

Y

120579

120579

120593

(b)




119909119901= minus119871 sin 120579 cos120593

119910119901= 119871 sin 120579

119911119901= minus119871 cos 120579 cos120593

(3)


120579 = 2

120579 tan120593 minus

119892

119871

sin 120579cos120593

+

1

119871

cos 120579cos120593

= minus120579

2

tan120593 minus119892

119871

cos 120579 sin120593

minus

1

119871

( sin 120579 sin120593 + 119910 cos120593)

(4)




1205960= radic

119892

119871

120577 =

119888

2119898

radic

119871

119892

(5)


120579 (119905) = 119860

1205960


0119905radic1 minus 120577

2) (6)



2

(7)


Δ119879 =

120587

1205960radic1 minus 120577

2 (8)






lowast =




119881MAX [rads] 1 1 2 2 2120587 2120587 2120587

119902UB [degree] 177 91 174 137 255 165 360








E1

W1

S1

E2

W2

S2

S3

X5

Y5

X3

Y3

X1

Y1

X6

Z6

X4

Z4

X2

Z2

X0

Z0


031

5

045

05

015

02

025

X

Z




119860119894

= [

119894minus1119877119894

119894minus1119901119894

0 1

]

=

[[[[[

[

cos 120579119894minus sin 120579

119894cos120572119894

sin 120579119894sin120572119894119886119894cos 120579119894

sin 120579119894

cos 120579119894cos120572119894minus cos 120579

119894sin120572119894119886119894sin 120579119894

0 sin120572119894

cos120572119894

119889119894

0 0 0 1

]]]]]

]

(9)



119894119879119895= 119860119894+1119860119894+2sdot sdot sdot 119860119895= [

119894119877119895

119894119901119895

0 1

] (10)







1205791= atan2 (119910w 119909w)

1205794= atan2 (radic1 minus 1198632 119863)



120583

2

(11)



119863 =

119909w2+ 119910w2+ (119911w minus 1198711)

2

minus 1198712

2minus 1198713

2

211987121198713

(12)


5 1205796




4) Then the


41198777= (01198774)

119879

(01198777) (13)


1205795= atan2 (4119877

7 [2 3]

41198777 [1 3]) plusmn 120587 (14)

1205796= atan2 (radic1 minus 4119877

7[3 3]241198777 [3 3]) (15)

1205797= atan2 (4119877

7 [3 2]

41198777 [3 1]) plusmn 120587 (16)


7at row 119903 and







119902119894(119905) = 119886

119894(119905 minus 119905119894)3

+ 119887119894(119905 minus 119905119894)2

+ 119888119894(119905 minus 119905119894) + 119889119894 (17)









Initial solution






Input shaping





Shaped trajectory





Time

Join

t ang

le

1205791120579n+1

q1(t) qn(t)

t1 t2 tn tn+1

h1 hn








120579119894 and time


120579119894ℎ119894+ 2

120579119894+1(ℎ119894+ ℎ119894+1) +

120579119894+2ℎ119894+1

= 6(

120579119894+2minus 120579119894+1

ℎ119894+1

minus

120579119894+1minus 120579119894

ℎ119894

)

(18)


1=120579119899+1= 0) we have a total



120579119899+1






minx 119865 =

119899

sum

119894=1

ℎ119894+ 119908119886int 1199022

119894119889119905


119894119895

10038161003816100381610038161003816le 119902

MAX119895


10038161003816100381610038161003816Vlowast119894119895

10038161003816100381610038161003816le VMAX119895



X

Y


X

Y



10038161003816100381610038161003816

120579

10038161003816100381610038161003816le 119869

MAX


119889119894(119871119895 119874) ge 119903

119871119895+ 119903119874


(19)


x = (ℎ1 ℎ

119899 120579119894119895 120579

119899119898)


(20)






119894(119871119895 119874)








02

Joint S1

01

Joint S2

0

2Joint E1

Time (s)

Join

t ang

le

minus2

minus2

minus1

0 05 1 15 2 25

0 05 1 15 2 25

0 05 1 15 2 25

(rad

)Jo

int a

ngle

(rad

)Jo

int a

ngle

(rad

)

(a) Joint angle

0

1

0

1

0

2

Join

t vel

ocity

minus2

minus1

minus1

Joint S1

Joint S2

Joint E1

Time (s)

0 05 1 15 2 25

0 05 1 15 2 25

0 05 1 15 2 25

(rad

s)

Join

t vel

ocity

(rad

s)

Join

t vel

ocity

(rad

s)

(b) Joint velocity







119904 =

120574119891minus 120574119904

119879

(21)






0 05 1 15 2 25 3 35

0

01

02

03

04

05

06

Time (s)

Velo

city

(ms

)

OriginalZVD shaped

minus01

minus02

minus03

minus04

(a) X direction

Velo

city

(ms

)

0

02

04

06

08

1

minus020 05 1 15 2 25 3 35

Time (s)

OriginalZVD shaped

(b) Y direction


0 05 1 15 2 25 3 35 4 45 5

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

120579

120601

Time (s)

(a) Without shaping

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)




7is the desired


01198777=[[

[


0 0 1

]]

]

(22)




X

Y

08 06 04 02 00

02

04

06

08

1



(a) Without shaping

X

Y

08 06 04 02 00

02

04

06

08

1





0 005

0

005

01

015

02

025

03

X

Y

minus025

minus005



(a) Without shaping


0 005

0

005

01

015

02

025

03

X

Y

minus025

minus005


(b) ZV2lin shaped






Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)

(a) Without shaping

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)



Center of container

X

Y

08 06 04 02 00

02

04

06

08

1



(a) Without shaping

X

0

02

04

06

08

1


Y








0 1 2 3 4 5

0

0005

001

0015

Time (s)

Liqu

id su

rface

disp

lace

men

t (m

)


minus0005

minus001


0 2 4 6 8 100

05

1

15

2

Frequency (Hz)

FFT

ampl

itude

23926

22949





0 1 2 3 4 5

0

002

004

006

Liqu

id su

rface

disp

lace

men

t (m

)

OriginalZV shaped

ZVD shaped

Time (s)

minus002

minus004

minus006


Y

(08 01 03)

(01 09 02)

(05 05 01)

X (03 minus08 02)




Disp

lace

men

t (m

m)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(a) Case 1D

ispla

cem

ent (

mm

)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(b) Case 2

100

120

140

160

180

200

220

240

Disp

lace

men

t (m

m)



0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD

(c) Case 3







1 (03 minus08 02) (08 01 03) minus1205872 02 (08 01 03) (01 09 02) 0 12058723 (01 09 02) (05 06 09) 1205872 0





1 1791 2197 25735 6123 762072 1783 2189 221 5929 731723 1156 1562 45997 9921 78431







References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












lowast =




119881MAX [rads] 1 1 2 2 2120587 2120587 2120587

119902UB [degree] 177 91 174 137 255 165 360








E1

W1

S1

E2

W2

S2

S3

X5

Y5

X3

Y3

X1

Y1

X6

Z6

X4

Z4

X2

Z2

X0

Z0


031

5

045

05

015

02

025

X

Z




119860119894

= [

119894minus1119877119894

119894minus1119901119894

0 1

]

=

[[[[[

[

cos 120579119894minus sin 120579

119894cos120572119894

sin 120579119894sin120572119894119886119894cos 120579119894

sin 120579119894

cos 120579119894cos120572119894minus cos 120579

119894sin120572119894119886119894sin 120579119894

0 sin120572119894

cos120572119894

119889119894

0 0 0 1

]]]]]

]

(9)



119894119879119895= 119860119894+1119860119894+2sdot sdot sdot 119860119895= [

119894119877119895

119894119901119895

0 1

] (10)







1205791= atan2 (119910w 119909w)

1205794= atan2 (radic1 minus 1198632 119863)



120583

2

(11)



119863 =

119909w2+ 119910w2+ (119911w minus 1198711)

2

minus 1198712

2minus 1198713

2

211987121198713

(12)


5 1205796




4) Then the


41198777= (01198774)

119879

(01198777) (13)


1205795= atan2 (4119877

7 [2 3]

41198777 [1 3]) plusmn 120587 (14)

1205796= atan2 (radic1 minus 4119877

7[3 3]241198777 [3 3]) (15)

1205797= atan2 (4119877

7 [3 2]

41198777 [3 1]) plusmn 120587 (16)


7at row 119903 and







119902119894(119905) = 119886

119894(119905 minus 119905119894)3

+ 119887119894(119905 minus 119905119894)2

+ 119888119894(119905 minus 119905119894) + 119889119894 (17)









Initial solution






Input shaping





Shaped trajectory





Time

Join

t ang

le

1205791120579n+1

q1(t) qn(t)

t1 t2 tn tn+1

h1 hn








120579119894 and time


120579119894ℎ119894+ 2

120579119894+1(ℎ119894+ ℎ119894+1) +

120579119894+2ℎ119894+1

= 6(

120579119894+2minus 120579119894+1

ℎ119894+1

minus

120579119894+1minus 120579119894

ℎ119894

)

(18)


1=120579119899+1= 0) we have a total



120579119899+1






minx 119865 =

119899

sum

119894=1

ℎ119894+ 119908119886int 1199022

119894119889119905


119894119895

10038161003816100381610038161003816le 119902

MAX119895


10038161003816100381610038161003816Vlowast119894119895

10038161003816100381610038161003816le VMAX119895



X

Y


X

Y



10038161003816100381610038161003816

120579

10038161003816100381610038161003816le 119869

MAX


119889119894(119871119895 119874) ge 119903

119871119895+ 119903119874


(19)


x = (ℎ1 ℎ

119899 120579119894119895 120579

119899119898)


(20)






119894(119871119895 119874)








02

Joint S1

01

Joint S2

0

2Joint E1

Time (s)

Join

t ang

le

minus2

minus2

minus1

0 05 1 15 2 25

0 05 1 15 2 25

0 05 1 15 2 25

(rad

)Jo

int a

ngle

(rad

)Jo

int a

ngle

(rad

)

(a) Joint angle

0

1

0

1

0

2

Join

t vel

ocity

minus2

minus1

minus1

Joint S1

Joint S2

Joint E1

Time (s)

0 05 1 15 2 25

0 05 1 15 2 25

0 05 1 15 2 25

(rad

s)

Join

t vel

ocity

(rad

s)

Join

t vel

ocity

(rad

s)

(b) Joint velocity







119904 =

120574119891minus 120574119904

119879

(21)






0 05 1 15 2 25 3 35

0

01

02

03

04

05

06

Time (s)

Velo

city

(ms

)

OriginalZVD shaped

minus01

minus02

minus03

minus04

(a) X direction

Velo

city

(ms

)

0

02

04

06

08

1

minus020 05 1 15 2 25 3 35

Time (s)

OriginalZVD shaped

(b) Y direction


0 05 1 15 2 25 3 35 4 45 5

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

120579

120601

Time (s)

(a) Without shaping

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)




7is the desired


01198777=[[

[


0 0 1

]]

]

(22)




X

Y

08 06 04 02 00

02

04

06

08

1



(a) Without shaping

X

Y

08 06 04 02 00

02

04

06

08

1





0 005

0

005

01

015

02

025

03

X

Y

minus025

minus005



(a) Without shaping


0 005

0

005

01

015

02

025

03

X

Y

minus025

minus005


(b) ZV2lin shaped






Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)

(a) Without shaping

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)



Center of container

X

Y

08 06 04 02 00

02

04

06

08

1



(a) Without shaping

X

0

02

04

06

08

1


Y








0 1 2 3 4 5

0

0005

001

0015

Time (s)

Liqu

id su

rface

disp

lace

men

t (m

)


minus0005

minus001


0 2 4 6 8 100

05

1

15

2

Frequency (Hz)

FFT

ampl

itude

23926

22949





0 1 2 3 4 5

0

002

004

006

Liqu

id su

rface

disp

lace

men

t (m

)

OriginalZV shaped

ZVD shaped

Time (s)

minus002

minus004

minus006


Y

(08 01 03)

(01 09 02)

(05 05 01)

X (03 minus08 02)




Disp

lace

men

t (m

m)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(a) Case 1D

ispla

cem

ent (

mm

)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(b) Case 2

100

120

140

160

180

200

220

240

Disp

lace

men

t (m

m)



0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD

(c) Case 3







1 (03 minus08 02) (08 01 03) minus1205872 02 (08 01 03) (01 09 02) 0 12058723 (01 09 02) (05 06 09) 1205872 0





1 1791 2197 25735 6123 762072 1783 2189 221 5929 731723 1156 1562 45997 9921 78431







References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119894119879119895= 119860119894+1119860119894+2sdot sdot sdot 119860119895= [

119894119877119895

119894119901119895

0 1

] (10)







1205791= atan2 (119910w 119909w)

1205794= atan2 (radic1 minus 1198632 119863)



120583

2

(11)



119863 =

119909w2+ 119910w2+ (119911w minus 1198711)

2

minus 1198712

2minus 1198713

2

211987121198713

(12)


5 1205796




4) Then the


41198777= (01198774)

119879

(01198777) (13)


1205795= atan2 (4119877

7 [2 3]

41198777 [1 3]) plusmn 120587 (14)

1205796= atan2 (radic1 minus 4119877

7[3 3]241198777 [3 3]) (15)

1205797= atan2 (4119877

7 [3 2]

41198777 [3 1]) plusmn 120587 (16)


7at row 119903 and







119902119894(119905) = 119886

119894(119905 minus 119905119894)3

+ 119887119894(119905 minus 119905119894)2

+ 119888119894(119905 minus 119905119894) + 119889119894 (17)









Initial solution






Input shaping





Shaped trajectory





Time

Join

t ang

le

1205791120579n+1

q1(t) qn(t)

t1 t2 tn tn+1

h1 hn








120579119894 and time


120579119894ℎ119894+ 2

120579119894+1(ℎ119894+ ℎ119894+1) +

120579119894+2ℎ119894+1

= 6(

120579119894+2minus 120579119894+1

ℎ119894+1

minus

120579119894+1minus 120579119894

ℎ119894

)

(18)


1=120579119899+1= 0) we have a total



120579119899+1






minx 119865 =

119899

sum

119894=1

ℎ119894+ 119908119886int 1199022

119894119889119905


119894119895

10038161003816100381610038161003816le 119902

MAX119895


10038161003816100381610038161003816Vlowast119894119895

10038161003816100381610038161003816le VMAX119895



X

Y


X

Y



10038161003816100381610038161003816

120579

10038161003816100381610038161003816le 119869

MAX


119889119894(119871119895 119874) ge 119903

119871119895+ 119903119874


(19)


x = (ℎ1 ℎ

119899 120579119894119895 120579

119899119898)


(20)






119894(119871119895 119874)








02

Joint S1

01

Joint S2

0

2Joint E1

Time (s)

Join

t ang

le

minus2

minus2

minus1

0 05 1 15 2 25

0 05 1 15 2 25

0 05 1 15 2 25

(rad

)Jo

int a

ngle

(rad

)Jo

int a

ngle

(rad

)

(a) Joint angle

0

1

0

1

0

2

Join

t vel

ocity

minus2

minus1

minus1

Joint S1

Joint S2

Joint E1

Time (s)

0 05 1 15 2 25

0 05 1 15 2 25

0 05 1 15 2 25

(rad

s)

Join

t vel

ocity

(rad

s)

Join

t vel

ocity

(rad

s)

(b) Joint velocity







119904 =

120574119891minus 120574119904

119879

(21)






0 05 1 15 2 25 3 35

0

01

02

03

04

05

06

Time (s)

Velo

city

(ms

)

OriginalZVD shaped

minus01

minus02

minus03

minus04

(a) X direction

Velo

city

(ms

)

0

02

04

06

08

1

minus020 05 1 15 2 25 3 35

Time (s)

OriginalZVD shaped

(b) Y direction


0 05 1 15 2 25 3 35 4 45 5

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

120579

120601

Time (s)

(a) Without shaping

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)




7is the desired


01198777=[[

[


0 0 1

]]

]

(22)




X

Y

08 06 04 02 00

02

04

06

08

1



(a) Without shaping

X

Y

08 06 04 02 00

02

04

06

08

1





0 005

0

005

01

015

02

025

03

X

Y

minus025

minus005



(a) Without shaping


0 005

0

005

01

015

02

025

03

X

Y

minus025

minus005


(b) ZV2lin shaped






Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)

(a) Without shaping

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)



Center of container

X

Y

08 06 04 02 00

02

04

06

08

1



(a) Without shaping

X

0

02

04

06

08

1


Y








0 1 2 3 4 5

0

0005

001

0015

Time (s)

Liqu

id su

rface

disp

lace

men

t (m

)


minus0005

minus001


0 2 4 6 8 100

05

1

15

2

Frequency (Hz)

FFT

ampl

itude

23926

22949





0 1 2 3 4 5

0

002

004

006

Liqu

id su

rface

disp

lace

men

t (m

)

OriginalZV shaped

ZVD shaped

Time (s)

minus002

minus004

minus006


Y

(08 01 03)

(01 09 02)

(05 05 01)

X (03 minus08 02)




Disp

lace

men

t (m

m)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(a) Case 1D

ispla

cem

ent (

mm

)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(b) Case 2

100

120

140

160

180

200

220

240

Disp

lace

men

t (m

m)



0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD

(c) Case 3







1 (03 minus08 02) (08 01 03) minus1205872 02 (08 01 03) (01 09 02) 0 12058723 (01 09 02) (05 06 09) 1205872 0





1 1791 2197 25735 6123 762072 1783 2189 221 5929 731723 1156 1562 45997 9921 78431







References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Initial solution






Input shaping





Shaped trajectory





Time

Join

t ang

le

1205791120579n+1

q1(t) qn(t)

t1 t2 tn tn+1

h1 hn








120579119894 and time


120579119894ℎ119894+ 2

120579119894+1(ℎ119894+ ℎ119894+1) +

120579119894+2ℎ119894+1

= 6(

120579119894+2minus 120579119894+1

ℎ119894+1

minus

120579119894+1minus 120579119894

ℎ119894

)

(18)


1=120579119899+1= 0) we have a total



120579119899+1






minx 119865 =

119899

sum

119894=1

ℎ119894+ 119908119886int 1199022

119894119889119905


119894119895

10038161003816100381610038161003816le 119902

MAX119895


10038161003816100381610038161003816Vlowast119894119895

10038161003816100381610038161003816le VMAX119895



X

Y


X

Y



10038161003816100381610038161003816

120579

10038161003816100381610038161003816le 119869

MAX


119889119894(119871119895 119874) ge 119903

119871119895+ 119903119874


(19)


x = (ℎ1 ℎ

119899 120579119894119895 120579

119899119898)


(20)






119894(119871119895 119874)








02

Joint S1

01

Joint S2

0

2Joint E1

Time (s)

Join

t ang

le

minus2

minus2

minus1

0 05 1 15 2 25

0 05 1 15 2 25

0 05 1 15 2 25

(rad

)Jo

int a

ngle

(rad

)Jo

int a

ngle

(rad

)

(a) Joint angle

0

1

0

1

0

2

Join

t vel

ocity

minus2

minus1

minus1

Joint S1

Joint S2

Joint E1

Time (s)

0 05 1 15 2 25

0 05 1 15 2 25

0 05 1 15 2 25

(rad

s)

Join

t vel

ocity

(rad

s)

Join

t vel

ocity

(rad

s)

(b) Joint velocity







119904 =

120574119891minus 120574119904

119879

(21)






0 05 1 15 2 25 3 35

0

01

02

03

04

05

06

Time (s)

Velo

city

(ms

)

OriginalZVD shaped

minus01

minus02

minus03

minus04

(a) X direction

Velo

city

(ms

)

0

02

04

06

08

1

minus020 05 1 15 2 25 3 35

Time (s)

OriginalZVD shaped

(b) Y direction


0 05 1 15 2 25 3 35 4 45 5

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

120579

120601

Time (s)

(a) Without shaping

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)




7is the desired


01198777=[[

[


0 0 1

]]

]

(22)




X

Y

08 06 04 02 00

02

04

06

08

1



(a) Without shaping

X

Y

08 06 04 02 00

02

04

06

08

1





0 005

0

005

01

015

02

025

03

X

Y

minus025

minus005



(a) Without shaping


0 005

0

005

01

015

02

025

03

X

Y

minus025

minus005


(b) ZV2lin shaped






Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)

(a) Without shaping

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)



Center of container

X

Y

08 06 04 02 00

02

04

06

08

1



(a) Without shaping

X

0

02

04

06

08

1


Y








0 1 2 3 4 5

0

0005

001

0015

Time (s)

Liqu

id su

rface

disp

lace

men

t (m

)


minus0005

minus001


0 2 4 6 8 100

05

1

15

2

Frequency (Hz)

FFT

ampl

itude

23926

22949





0 1 2 3 4 5

0

002

004

006

Liqu

id su

rface

disp

lace

men

t (m

)

OriginalZV shaped

ZVD shaped

Time (s)

minus002

minus004

minus006


Y

(08 01 03)

(01 09 02)

(05 05 01)

X (03 minus08 02)




Disp

lace

men

t (m

m)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(a) Case 1D

ispla

cem

ent (

mm

)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(b) Case 2

100

120

140

160

180

200

220

240

Disp

lace

men

t (m

m)



0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD

(c) Case 3







1 (03 minus08 02) (08 01 03) minus1205872 02 (08 01 03) (01 09 02) 0 12058723 (01 09 02) (05 06 09) 1205872 0





1 1791 2197 25735 6123 762072 1783 2189 221 5929 731723 1156 1562 45997 9921 78431







References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










X

Y


X

Y



10038161003816100381610038161003816

120579

10038161003816100381610038161003816le 119869

MAX


119889119894(119871119895 119874) ge 119903

119871119895+ 119903119874


(19)


x = (ℎ1 ℎ

119899 120579119894119895 120579

119899119898)


(20)






119894(119871119895 119874)








02

Joint S1

01

Joint S2

0

2Joint E1

Time (s)

Join

t ang

le

minus2

minus2

minus1

0 05 1 15 2 25

0 05 1 15 2 25

0 05 1 15 2 25

(rad

)Jo

int a

ngle

(rad

)Jo

int a

ngle

(rad

)

(a) Joint angle

0

1

0

1

0

2

Join

t vel

ocity

minus2

minus1

minus1

Joint S1

Joint S2

Joint E1

Time (s)

0 05 1 15 2 25

0 05 1 15 2 25

0 05 1 15 2 25

(rad

s)

Join

t vel

ocity

(rad

s)

Join

t vel

ocity

(rad

s)

(b) Joint velocity







119904 =

120574119891minus 120574119904

119879

(21)






0 05 1 15 2 25 3 35

0

01

02

03

04

05

06

Time (s)

Velo

city

(ms

)

OriginalZVD shaped

minus01

minus02

minus03

minus04

(a) X direction

Velo

city

(ms

)

0

02

04

06

08

1

minus020 05 1 15 2 25 3 35

Time (s)

OriginalZVD shaped

(b) Y direction


0 05 1 15 2 25 3 35 4 45 5

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

120579

120601

Time (s)

(a) Without shaping

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)




7is the desired


01198777=[[

[


0 0 1

]]

]

(22)




X

Y

08 06 04 02 00

02

04

06

08

1



(a) Without shaping

X

Y

08 06 04 02 00

02

04

06

08

1





0 005

0

005

01

015

02

025

03

X

Y

minus025

minus005



(a) Without shaping


0 005

0

005

01

015

02

025

03

X

Y

minus025

minus005


(b) ZV2lin shaped






Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)

(a) Without shaping

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)



Center of container

X

Y

08 06 04 02 00

02

04

06

08

1



(a) Without shaping

X

0

02

04

06

08

1


Y








0 1 2 3 4 5

0

0005

001

0015

Time (s)

Liqu

id su

rface

disp

lace

men

t (m

)


minus0005

minus001


0 2 4 6 8 100

05

1

15

2

Frequency (Hz)

FFT

ampl

itude

23926

22949





0 1 2 3 4 5

0

002

004

006

Liqu

id su

rface

disp

lace

men

t (m

)

OriginalZV shaped

ZVD shaped

Time (s)

minus002

minus004

minus006


Y

(08 01 03)

(01 09 02)

(05 05 01)

X (03 minus08 02)




Disp

lace

men

t (m

m)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(a) Case 1D

ispla

cem

ent (

mm

)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(b) Case 2

100

120

140

160

180

200

220

240

Disp

lace

men

t (m

m)



0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD

(c) Case 3







1 (03 minus08 02) (08 01 03) minus1205872 02 (08 01 03) (01 09 02) 0 12058723 (01 09 02) (05 06 09) 1205872 0





1 1791 2197 25735 6123 762072 1783 2189 221 5929 731723 1156 1562 45997 9921 78431







References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










02

Joint S1

01

Joint S2

0

2Joint E1

Time (s)

Join

t ang

le

minus2

minus2

minus1

0 05 1 15 2 25

0 05 1 15 2 25

0 05 1 15 2 25

(rad

)Jo

int a

ngle

(rad

)Jo

int a

ngle

(rad

)

(a) Joint angle

0

1

0

1

0

2

Join

t vel

ocity

minus2

minus1

minus1

Joint S1

Joint S2

Joint E1

Time (s)

0 05 1 15 2 25

0 05 1 15 2 25

0 05 1 15 2 25

(rad

s)

Join

t vel

ocity

(rad

s)

Join

t vel

ocity

(rad

s)

(b) Joint velocity







119904 =

120574119891minus 120574119904

119879

(21)






0 05 1 15 2 25 3 35

0

01

02

03

04

05

06

Time (s)

Velo

city

(ms

)

OriginalZVD shaped

minus01

minus02

minus03

minus04

(a) X direction

Velo

city

(ms

)

0

02

04

06

08

1

minus020 05 1 15 2 25 3 35

Time (s)

OriginalZVD shaped

(b) Y direction


0 05 1 15 2 25 3 35 4 45 5

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

120579

120601

Time (s)

(a) Without shaping

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)




7is the desired


01198777=[[

[


0 0 1

]]

]

(22)




X

Y

08 06 04 02 00

02

04

06

08

1



(a) Without shaping

X

Y

08 06 04 02 00

02

04

06

08

1





0 005

0

005

01

015

02

025

03

X

Y

minus025

minus005



(a) Without shaping


0 005

0

005

01

015

02

025

03

X

Y

minus025

minus005


(b) ZV2lin shaped






Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)

(a) Without shaping

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)



Center of container

X

Y

08 06 04 02 00

02

04

06

08

1



(a) Without shaping

X

0

02

04

06

08

1


Y








0 1 2 3 4 5

0

0005

001

0015

Time (s)

Liqu

id su

rface

disp

lace

men

t (m

)


minus0005

minus001


0 2 4 6 8 100

05

1

15

2

Frequency (Hz)

FFT

ampl

itude

23926

22949





0 1 2 3 4 5

0

002

004

006

Liqu

id su

rface

disp

lace

men

t (m

)

OriginalZV shaped

ZVD shaped

Time (s)

minus002

minus004

minus006


Y

(08 01 03)

(01 09 02)

(05 05 01)

X (03 minus08 02)




Disp

lace

men

t (m

m)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(a) Case 1D

ispla

cem

ent (

mm

)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(b) Case 2

100

120

140

160

180

200

220

240

Disp

lace

men

t (m

m)



0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD

(c) Case 3







1 (03 minus08 02) (08 01 03) minus1205872 02 (08 01 03) (01 09 02) 0 12058723 (01 09 02) (05 06 09) 1205872 0





1 1791 2197 25735 6123 762072 1783 2189 221 5929 731723 1156 1562 45997 9921 78431







References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 05 1 15 2 25 3 35

0

01

02

03

04

05

06

Time (s)

Velo

city

(ms

)

OriginalZVD shaped

minus01

minus02

minus03

minus04

(a) X direction

Velo

city

(ms

)

0

02

04

06

08

1

minus020 05 1 15 2 25 3 35

Time (s)

OriginalZVD shaped

(b) Y direction


0 05 1 15 2 25 3 35 4 45 5

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

120579

120601

Time (s)

(a) Without shaping

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)




7is the desired


01198777=[[

[


0 0 1

]]

]

(22)




X

Y

08 06 04 02 00

02

04

06

08

1



(a) Without shaping

X

Y

08 06 04 02 00

02

04

06

08

1





0 005

0

005

01

015

02

025

03

X

Y

minus025

minus005



(a) Without shaping


0 005

0

005

01

015

02

025

03

X

Y

minus025

minus005


(b) ZV2lin shaped






Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)

(a) Without shaping

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)



Center of container

X

Y

08 06 04 02 00

02

04

06

08

1



(a) Without shaping

X

0

02

04

06

08

1


Y








0 1 2 3 4 5

0

0005

001

0015

Time (s)

Liqu

id su

rface

disp

lace

men

t (m

)


minus0005

minus001


0 2 4 6 8 100

05

1

15

2

Frequency (Hz)

FFT

ampl

itude

23926

22949





0 1 2 3 4 5

0

002

004

006

Liqu

id su

rface

disp

lace

men

t (m

)

OriginalZV shaped

ZVD shaped

Time (s)

minus002

minus004

minus006


Y

(08 01 03)

(01 09 02)

(05 05 01)

X (03 minus08 02)




Disp

lace

men

t (m

m)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(a) Case 1D

ispla

cem

ent (

mm

)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(b) Case 2

100

120

140

160

180

200

220

240

Disp

lace

men

t (m

m)



0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD

(c) Case 3







1 (03 minus08 02) (08 01 03) minus1205872 02 (08 01 03) (01 09 02) 0 12058723 (01 09 02) (05 06 09) 1205872 0





1 1791 2197 25735 6123 762072 1783 2189 221 5929 731723 1156 1562 45997 9921 78431







References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










X

Y

08 06 04 02 00

02

04

06

08

1



(a) Without shaping

X

Y

08 06 04 02 00

02

04

06

08

1





0 005

0

005

01

015

02

025

03

X

Y

minus025

minus005



(a) Without shaping


0 005

0

005

01

015

02

025

03

X

Y

minus025

minus005


(b) ZV2lin shaped






Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)

(a) Without shaping

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)



Center of container

X

Y

08 06 04 02 00

02

04

06

08

1



(a) Without shaping

X

0

02

04

06

08

1


Y








0 1 2 3 4 5

0

0005

001

0015

Time (s)

Liqu

id su

rface

disp

lace

men

t (m

)


minus0005

minus001


0 2 4 6 8 100

05

1

15

2

Frequency (Hz)

FFT

ampl

itude

23926

22949





0 1 2 3 4 5

0

002

004

006

Liqu

id su

rface

disp

lace

men

t (m

)

OriginalZV shaped

ZVD shaped

Time (s)

minus002

minus004

minus006


Y

(08 01 03)

(01 09 02)

(05 05 01)

X (03 minus08 02)




Disp

lace

men

t (m

m)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(a) Case 1D

ispla

cem

ent (

mm

)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(b) Case 2

100

120

140

160

180

200

220

240

Disp

lace

men

t (m

m)



0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD

(c) Case 3







1 (03 minus08 02) (08 01 03) minus1205872 02 (08 01 03) (01 09 02) 0 12058723 (01 09 02) (05 06 09) 1205872 0





1 1791 2197 25735 6123 762072 1783 2189 221 5929 731723 1156 1562 45997 9921 78431







References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)

(a) Without shaping

Sway

angl

e (ra

d)

00102030405

minus01

minus02

minus03

minus04

minus05

0 05 1 15 2 25 3 35 4 45 5

120579

120601

Time (s)



Center of container

X

Y

08 06 04 02 00

02

04

06

08

1



(a) Without shaping

X

0

02

04

06

08

1


Y








0 1 2 3 4 5

0

0005

001

0015

Time (s)

Liqu

id su

rface

disp

lace

men

t (m

)


minus0005

minus001


0 2 4 6 8 100

05

1

15

2

Frequency (Hz)

FFT

ampl

itude

23926

22949





0 1 2 3 4 5

0

002

004

006

Liqu

id su

rface

disp

lace

men

t (m

)

OriginalZV shaped

ZVD shaped

Time (s)

minus002

minus004

minus006


Y

(08 01 03)

(01 09 02)

(05 05 01)

X (03 minus08 02)




Disp

lace

men

t (m

m)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(a) Case 1D

ispla

cem

ent (

mm

)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(b) Case 2

100

120

140

160

180

200

220

240

Disp

lace

men

t (m

m)



0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD

(c) Case 3







1 (03 minus08 02) (08 01 03) minus1205872 02 (08 01 03) (01 09 02) 0 12058723 (01 09 02) (05 06 09) 1205872 0





1 1791 2197 25735 6123 762072 1783 2189 221 5929 731723 1156 1562 45997 9921 78431







References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1 2 3 4 5

0

0005

001

0015

Time (s)

Liqu

id su

rface

disp

lace

men

t (m

)


minus0005

minus001


0 2 4 6 8 100

05

1

15

2

Frequency (Hz)

FFT

ampl

itude

23926

22949





0 1 2 3 4 5

0

002

004

006

Liqu

id su

rface

disp

lace

men

t (m

)

OriginalZV shaped

ZVD shaped

Time (s)

minus002

minus004

minus006


Y

(08 01 03)

(01 09 02)

(05 05 01)

X (03 minus08 02)




Disp

lace

men

t (m

m)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(a) Case 1D

ispla

cem

ent (

mm

)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(b) Case 2

100

120

140

160

180

200

220

240

Disp

lace

men

t (m

m)



0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD

(c) Case 3







1 (03 minus08 02) (08 01 03) minus1205872 02 (08 01 03) (01 09 02) 0 12058723 (01 09 02) (05 06 09) 1205872 0





1 1791 2197 25735 6123 762072 1783 2189 221 5929 731723 1156 1562 45997 9921 78431







References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Disp

lace

men

t (m

m)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(a) Case 1D

ispla

cem

ent (

mm

)

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD



(b) Case 2

100

120

140

160

180

200

220

240

Disp

lace

men

t (m

m)



0 1 2 3 4 5 6 7 8 9 10

Cubic

Time (s)

Cubic + ZVD

(c) Case 3







1 (03 minus08 02) (08 01 03) minus1205872 02 (08 01 03) (01 09 02) 0 12058723 (01 09 02) (05 06 09) 1205872 0





1 1791 2197 25735 6123 762072 1783 2189 221 5929 731723 1156 1562 45997 9921 78431







References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1 1791 2197 25735 6123 762072 1783 2189 221 5929 731723 1156 1562 45997 9921 78431







References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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