On preshaped reference inputs to reduce swingof suspended objects transported with robot

manipulators

GuÈ rsel Alõcõ*, Sadettin Kapucu, Sedat Baysec°

Gaziantep University, Faculty of Engineering, Department of Mechanical Engineering, TR-27310

Gaziantep, Turkey

Received 29 July 1999; accepted 17 September 1999

Abstract

One of the application areas of robot manipulators is pick-and-place operations wheresome objects may not be grasped by robot manipulators due to the task constraints. Such

objects, for example those simply suspended, need to be carried by a hook or a similardevice attached to the manipulator endpoint. This paper presents the results andimplications of a study into preshaped reference inputs to reduce the swing of suspended

objects at the end of a desired trajectory. Two reference input preshaping methods areconsidered; (i) superimposing a ramp function onto another function, and (ii) convolving asequence of impulses with a desired reference input to generate a shaped input. A

hydraulically actuated robot manipulator carrying a compound pendulum was employed asan experimental system to test the methods. Simulation and experimental results arepresented to demonstrate the feasibility of both methods. It is concluded that whilesuperimposing a ramp function onto another function such as a cycloid does not increase

the transportation time while reducing the swing, whereas the latter, convolving a sequenceof impulses with a cycloid, increases the transportation time. In addition to this, therobustness of the ®rst method to the uncertainties in the natural frequency of the suspended

object is better than that of the second method. The simulation and experimental resultsprove that by properly preshaping the reference input of a robot manipulator, a swing-freestop is obtainable. The proposed approaches are simple and easy to implement. 7 2000

Elsevier Science Ltd. All rights reserved.

0957-4158/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved.

PII: S0957 -4158 (99)00092 -6

Mechatronics 10 (2000) 609±626

* Corresponding author. Tel.: +90-342-360-1200 ext. 2508.

E-mail address: gursel@gantep.edu.tr (G. Alõcõ).

1. Introduction

In most robotic applications, the load carried by a robot manipulator is held®rmly and not allowed to move relative to the gripper. In such a case, the loadmay be assumed to be a part of the tip-most link. Many researchers consider theload like this and assume it to do the same motion as the tip-most link of themanipulator, but in some cases, the load may be able to move or swing, and thusdisplays a motion separate from the manipulator. Such applications can betypi®ed by an object suspended from the manipulator end-point via a hook-likedevice [1,11], the transport of large objects in a factory environment by the use ofa bridge crane by which the object is raised, transported and lowered on a targetlocation [2], and a molten-metal ®lled container carried by a manipulator in afoundry where the molten-metal should not splash at the end of thetransportation where pouring the metal into the molds takes place. The transportof such objects generally results in undesired swing at the end of a move.

In order to stop a suspended object in a swing-free state, Starr has suggested atrajectory consisting of an acceleration part [1], where the suspension point isaccelerated stepwise until half the desired transportation velocity is reached andthen is further stepwise accelerated to the speci®ed ®nal velocity which is attainedat a point roughly at one-fourth of the transportation velocity times the period ofnatural oscillation of the suspended object, and a deceleration part, where thesame process is applied in reverse. Later, Strip has reported on the swing-freetransportation of suspended objects that the suspension point has a symmetricalacceleration and deceleration pro®les [2], hence the deceleration starting at adistance from its goal equal to the distance traveled while accelerating. Thisdistance is reported to be equal to one-half of the acceleration of the suspensionpoint times the square of the period of natural oscillation of the object. In aprevious study by the authors of this paper [11], two other methods; (i) adjustingtransportation time, thus stopping the manipulator at the instant when the objectcompletes one or more number of full cycle(s); and (ii) adjusting the traveling timeof each section of a three-piece continuous trajectory provided that a giventransportation time is unchanged, have been used to demonstrate the swing-freetransportation of suspended objects with robot manipulators. The preshapingmethods evaluated in this paper are more general and practical than the twomethods proposed in the previous work mentioned above.

In this study, two methods based on preshaping an appropriate trajectory forthe suspension point are examined in order to eliminate swing at the end of anymove. Trajectories employed are di�erent than those reported before [1±5,11,14].It is therefore believed that this study contributes to the e�orts to eliminateswinging of suspended objects. We assume that the compliance existing in thetransmission and structural elements does not cause considerable vibration of themanipulator end-point during and at the end of the move, and the suspensionpoint follows a pre-determined path with a certain velocity and acceleration. Thisis practically veri®ed by the experimental results presented in Section 4, where thetranslating link of the manipulator follows a commanded trajectory with a good

G. Alõcõ et al. / Mechatronics 10 (2000) 609±626610

®t. To demonstrate the feasibility of both methods, we consider two trajectoriesbased on a cycloidal motion which is commonly used as a high-speed cam pro®le,continuous in the ®rst and second derivatives throughout one cycle. This does notintroduce any high frequency inputs to the system and hence reduces unwanteddynamic e�ects. A ramp function superimposed onto a cycloid is proposed for the®rst method, and the same full cycloid is convolved with a sequence of twoimpulses to obtain a shaped input for the second method. The two methods aresimilar to each other in respect to dividing the reference input into two pieceswhich generate two equal and out of phase responses. Input preshaping based onconvolving the desired reference input with a sequence of impulses is a method ofreducing residual vibrations in computer-controlled machines [3,6,12,14]. Thismethod requires a perfect knowledge of the system parameters such as naturalfrequency and damping ratio to reduce residual vibration to zero. The moreimpulses are used, the more robust the system becomes to system parametervariations but at a penalty of greater response time. Meckl and Seering [13], havedeveloped a set of shaped force pro®les to reduce the residual vibrations ofvelocity limited dynamic systems. The force pro®les are constructed from a versine(1-cosine) function, which has no discontinuity in slope at the beginning and atthe end, and its harmonics with coe�cients chosen to minimize the energy of theresulting function near the system natural frequency. This is accomplished byemploying an optimization technique.

The method of superimposing a ramp onto another function does not restrictthe traveling time. The amplitude of the ramp and the other function aredetermined such that a total distance to be traveled and the transportation timeare satis®ed. The ramp function can be superimposed onto any mathematicallydescribed function in order to give an initial velocity to the system, whichgenerates a swing equal in magnitude, but out of phase to the swing imposed bythe main trajectory. On the other hand, the method of convolving a sequence ofimpulses with a desired reference trajectory increases the transportation time inorder to reduce the swing. An object suspended at the end of a hydraulicallyactuated robot manipulator by a hook-like device is considered and the equationsof motion for the translating link of the manipulator are solved on a digitalcomputer for the desired trajectory of the suspension point. The servo valve andactuator dynamics are considered in the simulations. Simulation results have beenveri®ed on the robot manipulator whose translational link carried a compoundpendulum. Simulation and experimental results reveal that both methods areapplicable and easy to implement on a real system.

2. Modelling of system

The experimental system used in this study consists of a single link hydraulicallyactuated robot manipulator transporting a simply suspended object, which isattached to the manipulator end-point by a hook or a similar device. A schematic

G. Alõcõ et al. / Mechatronics 10 (2000) 609±626 611

representation of the system is depicted in Fig. 1, where O is the point ofsuspension, x is the horizontal displacement of point O, y is the angulardisplacement of the object with respect to the vertical, M is the mass of thetranslating link, m is the mass of the object, IG is the mass moment of inertia ofthe suspended object about its center of gravity, r is the radial distance from thecenter of gravity G to the point of suspension O. The equations of motion for thesystem, which consists of the translating link and simply suspended object, areobtained as [7]:

�M�m� �x� �mr cos y��yÿmr_y2

sin y� c _x � Fa �1�

�IG �mr2��y� �mr cos y� �x�mgr sin y � 0 �2�

where Fa � P1A1 ÿ P2A2, the force exerted by the actuator. P1 and P2 are thepressures in the actuator, and are the functions of supply pressure, coe�cient ofleakage ¯ow, ¯ow gain of spool stage, area of the piston, actuator speed andspool displacement [9], A1 and A2 are the cross-sectional areas of the piston. Notethat Eqs. (1) and (2) are coupled nonlinear di�erential equations which can belinearized by substituting sin y=y and cos y=1 into them with an error less than1%, if y<5.58 of its motion [8]. They, then, become:

�M�m� �x� �mr��yÿ �mr_y2�y� c _x � Fa �3�

�IG �mr2��y� �mr� �x�mgry � 0 �4�

Fig. 1. Schematic representation of a translating link carrying a simply suspended object via a hook.

G. Alõcõ et al. / Mechatronics 10 (2000) 609±626612

The swing of the suspended object is described by Eq. (4), where the accelerationxÈ of the point of suspension O is the forcing function which generates the swing.If it is possible to describe xÈ mathematically, the solution for Eq. (4) will beobtained. So, the motion of the suspended object is a simple-periodic motionwhile the point of suspension O is accelerating with xÈ and the natural frequency is

on ����������mgr

I0

r�5�

where I0 is the mass moment of inertia of the swinging object with respect to anaxis through O.

When a ramp function is used as a part of a desired trajectory, it becomesdi�cult to describe the acceleration of the ramp function at the beginning and endof the trajectory, where it becomes +1 and ÿ1, respectively. Thus, it becomesproblematic to solve Eq. (4) for a ramp input. To overcome this di�culty, wede®ne an equivalent system moving under the desired displacement y. Althoughthe suspended object depicted in Fig. 1, and the equivalent system shown in Fig. 2are di�erent physically, they are represented by the same mathematical model.Such systems are called equivalent or analogous systems. In this study, we makeuse of the mathematical model of the spring-mass system shown in Fig. 2,composed of an equivalent mass me moving in coordinate x and an equivalentsti�ness ke. It is:

�x� o2nx � o2

ny �6�where

o2n �

ke

me

The problem here is to plan a reference input that will move the translating linkof the manipulator, which carries the suspended object, from one operating pointto another in the shortest possible time with zero swing at the end of the move.By preshaping the reference input y, a swing-free move is obtainable. Note thatthe input is the displacement y, and the output is the angular displacement y ofthe suspended object. The pressures P1 and P2 are in quadratic forms and aregiven in [9]. The natural frequency on is the natural frequency of the suspendedobject, given by Eq. (5).

Fig. 2. An equivalent simple mass-spring system representing the swing of the simply suspended object.

G. Alõcõ et al. / Mechatronics 10 (2000) 609±626 613

3. Elimination of swing

The translating link of the manipulator is desired to follow the input y. A fullcycloid plus a ramp function, and a full cycloid convolved with a sequence of twoimpulses are the two displacement trajectories for the pre-determined trajectory ofthe translating link.

3.1. Superimposing a ramp function onto another function

The reference input for the translating link consists of two functions. The totaldistance to be covered from the beginning to the end of a move within a speci®edmove time is the sum of the distances to be traveled by each of the two functionswithin the same duration. By adjusting the excursion distance of each function,the swing of the suspended object can be eliminated provided that the speci®edmove time and the total distance are unchanged. Each component of the referenceinput for the translating link creates a swing of equal amplitude, out of phasesuch that they cancel each other and no swing results.

A full cycloidal motion pro®le whose continuous derivatives at the beginningand at the end of a move making it quite preferable is expressed as;

y1 � Y1

2p�2Rtÿ sin�2Rt�� �7�

where Y1 is the maximum distance to be traveled, t is time into motion, t is thetraveling time, and R=p/t.

A ramp input is given by

y2 � Y2

tt �8�

where Y2 is the maximum distance to be traveled at the end of the traveling timet.

As stated before, the corresponding acceleration pro®le of a ramp input hasdiscontinuities at the beginning and at the end of the motion, and therefore it isdi�cult to describe it mathematically. To overcome such a problem, an analogousspring-mass system shown in Fig. 2 is employed to determine the amplitudes ofthe ramp and the cycloidal functions. For zero initial conditions, the equation ofmotion for the analogous spring-mass system given by Eq. (6) is solved for theramp and cycloidal inputs de®ned by Eqs. (7) and (8) as:

x � Y14R3

pon�o2n ÿ 4R2� sin�ont� � Y1

2p

"2Rt�

o2

n

4R2 ÿ o2n

!sin�2Rt�

#

� Y2

t

�tÿ sin�ont�

on

� �9�

G. Alõcõ et al. / Mechatronics 10 (2000) 609±626614

This solution is valid until t=t. After that time, y � Y � Y1 � Y2, and thesolution of Eq. (6) becomes:

x � A sin on�tÿ t� � B cos on�tÿ t� � Y �10�The arbitrary constants A and B are evaluated from the ®nal value of Eq. (9) andits time derivative at t=t. They are found as:

A �"

4Y1R3

pon�o2n ÿ 4R2� ÿ

Y2

ton

#cos�ont� � Y

ton

ÿ Y1on

t�o2n ÿ 4R2�

B �"

4Y1R3

pon�o2n ÿ 4R2� ÿ

Y2

ton

#sin�ont� �11�

It is necessary to set A and B to zero simultaneously to have no swing after t> t.This yields the following solution for the distances Y1 and Y2

Y1 � Y

"1ÿ

�tn

t

�2#

Y2 � Y

�tn

t

�2

�12�

where tn is the period of natural oscillation or swing. The variation of Y1 and Y2

with the traveling time t, which results with the swing-free displacement of asystem whose tn arbitrarily taken as 1.4 s, is given in Fig. 3. When the travelingtime is equal to the period of natural oscillations of the system, the desiredtrajectory consists only of the ramp input. When it is less than the natural period,the amplitude of the cycloid part of the trajectory is negative.

Eqs. (3) and (4) are solved on a digital computer for the cycloidal input plus theramp input with the distances determined from Eq. (12), using the fourth and®fth-order Runge±Kutta numerical integration algorithms. The period ofoscillation of the compound pendulum considered is taken as 1.4 s or the naturalfrequency 4.5 rad/s which is the same as that of the experimental system describedin the next section. The distance Y to be covered and the traveling time t arearbitrarily taken as 0.3 m and 1.8 s, respectively, in the simulation. The same datais used in Section 4 for experimental veri®cation. The simulation result is depictedin Fig. 4 where the shaped input y, the response of the link x, and the resultingswing y of the suspended object are given together. When the distances Y1 and Y2

are provided without considering Eq. (12), the suspended object keeps swinging atthe end of the move, as seen in Fig. 5. As given in Eq. (12), Y1 and Y2 are thefunction of the natural frequency which is the identity of a dynamic system.Simulations are, therefore, carried out in order to demonstrate the robustness ofthis method to uncertainties in the natural frequency of the system. The maximum

G. Alõcõ et al. / Mechatronics 10 (2000) 609±626 615

Fig. 3. Variation of Y1 and Y2 with the traveling time for a unit swing-free displacement of a system

having a natural frequency of swing arbitrarily 4.5 rad/s.

Fig. 4. Simulated swing-free motion of the suspended object having a natural frequency of 4.5 rad/s for

Y1=0.1185 m and Y2=0.1815 m.

G. Alõcõ et al. / Mechatronics 10 (2000) 609±626616

amplitude of the residual swing, after a movement of unit step input applied for tarbitrarily 1.4 s, vs the ratio of o 'n/on is plotted in Fig. 6, where o 'n is the actualnatural frequency of the system. Note that this method can move the systemwithout causing swing when the system natural frequency is exact and with verylittle swing when the natural frequency is not known exactly. If the amplitude ofthe residual swing for a simple swinging system is less than 5%, it is widelyaccepted that the system is in a swing-free state after following a trajectory.

Fig. 6. Simulated maximum amplitude of the residual swing vs the ratio of o 'n/on for t=1.4 s.

Fig. 5. Simulated swinging motion of the suspended object having the natural frequency of 4.5 rad/s

for arbitrarily selected distances of Y1=Y2=0.15 m.

G. Alõcõ et al. / Mechatronics 10 (2000) 609±626 617

3.2. Impulse sequence based reference input preshaping

This method, which involves convolving a sequence of impulses with the maindesired trajectory in order to produce a shaped reference input, has been proposedby Singer and Seering. A full account of this method is given in [6] and [12]. It isbased on the transient response of a second order system with the naturalfrequency of on, and the expected damping ratio of z to an impulse input. Thetransient response c(t ) is expressed as:

c�t� ��A

on�������������1ÿ z2

p eÿzon�tÿt0��

sin�on

�������������1ÿ z2

q�tÿ t0�� �13�

where A is the amplitude of the position impulse command, t is time, t0 is thetime of the impulse input. Since any arbitrary function can be formed from asequence of impulses, the impulse sequence can be used to reduce the vibration ofdynamic systems or the swing of suspended objects under arbitrary trajectories.This superposition is accomplished by convolving any desired trajectory with asequence of impulses in order to yield the shortest actual system input. Thisoperation, therefore, becomes a pre®lter for any input to be given to the system.The time penalty resulting from pre®ltering the input equals to the length of theimpulse sequence.

The amplitudes and the durations of the impulses are determined such that thesystem moves without swing after the motion has ended. When a sequence of twoimpulses is used to reduce the residual vibration of a dynamic system, thefollowing impulse durations and amplitudes are found [6]:

t1 � 0, t2 � p

on

�������������1ÿ z2

pA1 � 1

1� K, A2 � K

1� K

where

K � e

ÿzp��������1ÿz2p

:

Note that the amplitudes of the two impulses are normalised so that they sum tounity. These are the shortest time-duration sequences that eliminate residualvibrations, assuming that only positive amplitudes are used. It has been reported[12] that the response is extremely sensitive to uncertainties in the naturalfrequency and damping ratio of the system. Some additional constraints can beadded to increase robustness against the uncertainties. This necessitates the use ofmore than two impulses at a penalty of increasing the move time [12], which isnot very desirable.

Recalling that for the system under consideration has no damping, that is,

G. Alõcõ et al. / Mechatronics 10 (2000) 609±626618

z=0.0. The durations and amplitudes of the impulses are found as:

t1 � 0, t2 � pon

A1 � 1

2, A2 � 1

2

where K=1. Note that the time t2 of the second impulse is a half of the period ofnatural oscillation of the system and the duration of the motion is lengthened byt2. This sequence of two impulses is convolved with any desired trajectory. In thisstudy, a cycloidal function which has continuous ®rst and second derivatives atthe beginning and at the end of a move is chosen as the desired trajectory. It isimportant to note that the preshaped trajectory does not contain impulses oncethe convolution is performed. The convolution of any desired trajectory with someimpulse sequence results in a trajectory that has the same vibration or swingreducing e�ects as the impulse sequence.

For t=1.1 s, Y = 0.3 m and on=4.5 rad/s, the duration of the move isincreased by t2=0.7 s. Note that after the convolution operation the total time oftravel becomes 1.8 s which is the same time for the method of superimposing aramp function onto another function. By using the preshaped trajectory consistingof a full cycloid convolved with a sequence of two impulses, the di�erentialequation of motion given by Eqs. (3) and (4) are solved on a digital computeragain by using the fourth and ®fth-order Runge±Kutta numerical integrationalgorithms. The solution obtained is depicted in Fig. 7, where it is clearly seenthat there is no swing when t> t.

When the desired cycloidal trajectory is not convolved with the impulses, asgiven in Fig. 8, the suspended object keeps swinging at the end of the move.

The maximum amplitude of the residual swing, after a desired trajectory of aunit step is applied for t=1.4 s, is plotted against o 'n/on in Fig. 9 which indicatesthat this method is also robust to uncertainties in the natural frequency of thesystem. The simulation results given in Fig. 6 and in Fig. 9 are depicted togetherin Fig. 10 in order to compare the robustness of both methods for the samedistance of Y = 0.3 m and the traveling time of t=1.4 s. Note the ®rst methodpresented in the previous subsection seems more robust to uncertainties in thenatural frequency and does not lengthen the traveling time to accomplish a giventrajectory.

4. Experimental veri®cation

The experimental set-up shown in Fig. 11 consists of a hydraulically actuatedmanipulator of Stanford type, a compound pendulum attached to the tip of thetranslating link comprising the dynamic load, a conductive plastic servopotentiometer to measure the swing of the suspended object, and hardware to

G. Alõcõ et al. / Mechatronics 10 (2000) 609±626 619

Fig. 7. Simulated swing-free motion of the suspended object for the cycloidal function convolved with a

sequence of two impulses.

Fig. 8. Simulated swinging motion of the suspended object for the cycloidal function.

G. Alõcõ et al. / Mechatronics 10 (2000) 609±626620

control and command the manipulator, and read the potentiometer output. Themanipulator links are controlled by Bosch regulator valves of 0811 404 028 type[10] in a closed loop fashion with rotary potentiometer feedback to obtain thedesired position of the translating link. The block diagram of the control strategyis shown in Fig. 12.

The object of the experimental work has been to illustrate the e�ectiveness androbustness of the methods in preventing the swing of suspended objects. Thetranslational link of the manipulator was kept parallel to the ground and wasgiven the preshaped trajectories described in Section 3.

Fig. 9. Simulated maximum amplitude of the residual swing vs the ratio of o 'n/on for t=1.4 s.

Fig. 10. Comparison of simulated maximum amplitude of the residual swing vs the ratio of o 'n/on for

t=1.4 s. Method 1 is the superposition of a ramp and cycloid, Method 2 is the convolution of a

cycloid with two impulses.

G. Alõcõ et al. / Mechatronics 10 (2000) 609±626 621

4.1. Experimental results

Two sets of experiments have been conducted. The ®rst set is for the method ofsuperimposing a ramp function onto a full cycloid and the second is for that ofconvolving a sequence of two impulses with a full cycloid. In all the results presentedin this section, the shaped input y, the resulting movement x of the point ofsuspension O, and the resulting experimentally measured and simulated swing y ofthe suspended object are shown in the same ®gure for comparison of simulationand experimental results. Fig. 13 shows the experimental and simulation resultstogether when the distances Y1 and Y2 are calculated from Eq. (12), for t=1.8 s,on=4.5 rad/s, and Y= 0.3 m. Note that the data used for this result is the samedata for the simulation result depicted in Fig. 4. Fig. 14 shows the results whenthe distances are taken arbitrarily as Y1=0.15 m and Y2=0.15 m. The simulationand experimental results given in the same ®gures indicate that there is a closecorrespondence between the results.

The experimental result shown in Fig. 15 is for the case when the cycloidalinput is convolved with a sequence of two impulses. The data used for this resultis the same data for the simulation result depicted in Fig. 7. Fig. 16 shows theexperimental result when the cycloidal input is not convolved with the impulsesfor t=1.8 s, and Y=0.3 m.

Fig. 11. Con®guration of the robot manipulator and experimental set-up.

Fig. 12. Control strategy for the position control of the translating link of the robot manipulator

shown in Fig. 11.

G. Alõcõ et al. / Mechatronics 10 (2000) 609±626622

The experimental results corresponding to the simulation results given in Fig. 10are also taken to verify the robustness of both methods to uncertainties in thenatural frequency of the system. These experimental and simulation results areshown together in Fig. 17. The close correspondence between them demonstratethat both methods perform well on a real system as shown by simulation.

Fig. 13. Experimental and simulated swing-free motion of the suspended object for Y1=0.1185 m

Y2=0.1815 m.

Fig. 14. Experimental and simulated swinging motion of the suspended object for Y1=Y2=0.15 m.

G. Alõcõ et al. / Mechatronics 10 (2000) 609±626 623

5. Conclusions

The simulation and experimental results show that it is possible to obtain aswing-free motion at the end of a move by employing the method ofsuperimposing a ramp function onto another function, and the method ofconvolving a sequence of two impulses with a desired reference input to generate ashaped input. While the ®rst method does not limit or increase the move time toeliminate swing at the end of the move, the latter lengthens the move time andnecessitates more impulses in order to be robust to the natural frequencyuncertainties. It is shown that both preshaping methods are simple and easy to

Fig. 15. Experimental and simulated swing-free motion of the suspended object for the cycloidal

function convolved with a sequence of two impulses.

Fig. 16. Experimental and simulated swinging motion of the suspended object for the cycloidal

function.

G. Alõcõ et al. / Mechatronics 10 (2000) 609±626624

implement, robust to uncertainties in the natural frequency of the swing system,and can be considered as two versatile ways to determine a trajectory resulting inno swing at the end of a move. It is, therefore, concluded that a swing-free stopcan be obtained by suitably preshaping the reference input for a robotmanipulator.

References

[1] Starr GP. Swing-free transport of suspended objects with a path controlled robot manipulator.

ASME J Dyn Syst Meas Contr 1985;107(1):97±100.

[2] Strip DR. Swing-free transport of suspended objects: a general treatment. IEEE Transactions on

Robotics and Automation 1989;5(2):234±6.

[3] Singhose WE, Singer NC. E�ects of input shaping on two dimensional trajectory following. IEEE

Transactions on Robotics and Automation 1996;12(6):881±7.

[4] McConnell KG, Bouton CE. Noise and vibration control with robust vibration suppression.

Sound and Vibration 1997;31(6):24±6.

[5] Jones JR. A comparison of dynamic performance of tuned and non-tuned multiharmonic cams.

In: Proceedings of the Second IFToMM International Symposium on Linkages and Computer

Aided Design Methods, Volume III-2, Bucharest, Romania, 1977. p. 553±64.

[6] Singer NC, Seering WP. Preshaping command inputs to reduce system vibration. ASME J Dyn

Syst Meas Contr 1990;112:76±82.

[7] Baysec° B, Jones JR. A generalised approach for the modelling of articulated open chain planar

linkages. Robotica 1997;15:523±31.

[8] Steidel RF. An introduction to mechanical vibrations, 2nd ed. Wiley NY, USA, 1979.

[9] Baysec° S, Jones JR. An improved model of an electrohydraulic servo-valve. In: Proceedings of the

Seventh IFToMM Congress, Vol. 3, Sevilla, Spain, 1987. p. 1489±94.

[10] Bosch Regulator valves, Robert Bosch GmbH, Germany, 761302, 1987.

Fig. 17. Comparison of experimental and simulated maximum amplitude of the residual swing vs the

ratio of o 'n/on for t=1.4 s.

G. Alõcõ et al. / Mechatronics 10 (2000) 609±626 625

[11] Alici G, Kapucu S, Baysec° S. Swing-free transportation of suspended objects with robot

manipulators. Robotica Vol 17: p. 513±521 1999.

[12] Singer NC. Residual vibration reduction in computer controlled machines. Ph.D. thesis,

Massachusetts Institute of Technology, January, 1989.

[13] Meckl PH, Seering WP. Controlling velocity-limited systems to reduce residual vibration. In:

Proceedings of the 1988 IEEE International Conference on Robotics and Automation, April,

Philadelphia, USA, 1988. p. 1428±33.

[14] Cuccio A, Garziera R. Vibration control input-laws in point-to-point motion: theory and

experiments. Mechanisms and Machines Theory 1998;33(4):341±9.

G. Alõcõ et al. / Mechatronics 10 (2000) 609±626626