Mathematical modelling and fexible robot manipulator
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Mathematical modelling and dynamic response of a multi-straight-line path tracing flexible robot manipulatorwith rotating-prismatic jointMete KalyoncuDepartment of Mechanical Engineering,Faculty of Engineering and Architecture,University of Selc ¸uk,Alaeddin Keykubat Campus,42079Konya,TurkeyReceived 1October 2005;received in revised form 1February 2007;accepted 6February 2007Available online 7April 2007AbstractIn this study,mathematical modelling and dynamic response of a flexible robot manipulator with rotating-prismatic joint are investigated.The tip end of the flexible robot manipulator traces a multi-straight-line path under the action of an external driving torque and an axial force.Considered robot manipulator consists of a rotating prismatic joint and a sliding flexible arm with a tip mass.Flexible arm is assumed to be an Euler–Bernoulli beam carrying an end-mass.Equa-tions of motion of the flexible manipulator are obtained by using Lagrange’s equation of motion.Effect of rotary inertia,axial shortening and gravitation is considered in the analysis.Equations of motion are solved by using fourth order Runge–Kutta method.Numerical simulations obtained by using a developed computer program are presented and phys-ical trend of the results are discussed.Ó2007Elsevier Inc.All rights reserved.Keywords:Flexible robot manipulator;Dynamic response;Straight-line path;Rotating-prismatic joint;Mathematical modelling1.IntroductionResearch on the dynamic modelling of flexible manipulators has received increased attention in the last dec-ades due to their several advantages over rigid manipulators.One of the major problems related to manipu-lators and robots is the excessive weight of the members.For accurate position control,the members are designed for rigidity constraint resulting with heavy robots or manipulators.Heavy arm means heavy actua-tor,heavy actuator means heavy joint.Every part large in size means higher cost.Heavy arms need more energy and driving power,which is another disadvantage.For some applications,high-velocity requirement needs low-inertia values in order to prevent from undesired dynamic effects.Increasing demand for high-speed performance and low-energy consumption of robotic systems coupled with needs of limited space applications have necessitated the design of light-weight manipulators.It is well known that the demand for increased0307-904X/$-see front matter Ó2007Elsevier Inc.All rights reserved.doi:10.1016/j.apm.2007.02.032E-mail address:metekalyoncu@Available online at Applied Mathematical Modelling 32(2008)1087–1098/locate/apm1088M.Kalyoncu/Applied Mathematical Modelling32(2008)1087–1098productivity by robots can be partly met by the use of lighter robots operating at high-speed and consuming less energy,which may lead to a reduction in the stiffness of the manipulator structure.Linkflexibility is a consequence of the lightweight constructional feature in manipulator arms that are designed to operate at high-speeds.This would result in an increase in elastic deflections and poor performance due to the effect of mechanical vibration in the links and make the robot control more difficult.Thus,trajectory control of a robotic manipulator system has been an important research area in the last decade.Flexible manipulators undergo a combination of rigid andflexible motions.Because of the interaction of these motions,the resulting dynamic equations offlexible manipulators are highly complex and,in turn,the control task becomes more challenging compared to that for rigid robots.Therefore,afirst step towards designing an efficient control strategy for these manipulators must be aimed at developing accurate dynamic models that can characterize theflexibility of the links.Most of the investigations on the dynamics of robot manipulators with elastic arms consider manipulators with revolute joints[1–5].Dynamics of an elastic arm sliding in a prismatic joint is an important problem in many engineering applications.Elastic arm sliding in a prismatic joint can be seen in many robot applications, telescopic members of loading vehicles,space craft antenna,magnetic tape drivers,printers,flexible transmis-sion lines,band saws and weaving mechanisms[6,7].Flexible members sliding in prismatic joints are known to produce considerable mathematical difficulty in the dynamic modelling of such systems.The problem becomes even more difficult if the beam is translating through a rotating prismatic joint.It has become evident that a reliable dynamic model for a translating and rotating beam that accounts for the interaction between rigid and flexible body motions is highly demanded.Such a dynamic model is crucial to the design,performance eval-uation,and control of light-weight,high-speed,and high-precision applications.Wang and Wei[8]modeled aflexible robot arm by a moving slender prismatic beam.The vibration analysis was based on a Galerkin approximation scheme using time dependent basis functions.Numerical results obtained for some typical problems were presented.The effect of extending and contracting motions on the vibratory motion was discussed.Kane et al.[9]studied the vibrations of a cantilever beam mounted on a moving foundation considering the effect of centrifugal and Coriolis forces.In a similar work,Gaultier and Cleghorn[10]usedfinite element method to analyze the vibration of a spatially translating and rotating beam to model elastic link manipulators.The internal and external damping effects were also included in the model. Pan et al.[11]analyzed the vibration offlexible manipulators with prismatic joint.The prismatic joint was modeled as a telescopic manipulator composed of two elastic links.Equations of motion were obtained by Lagrange’s equation of motion.The boundary conditions were introduced into the formulation by Lagrange multipliers.The effect of axial shortening caused by the orthogonal deformations was also taken into consid-eration.Pan et al.[12]used an experimental scheme in order to validate the proposed dynamic model.Yuh and Young[13]analyzed the dynamics of a beam experiencing a combination of rotational and translational motions.An approximation scheme was developed by using assumed modes method.The validity of the approximate model was evaluated by a series of experimental work.Dynamic response of the elastic beam undergoing various motions was investigated by computer simulation.Tadikonda and Baruh[14]analyzed the vibration and control of an elastic arm with an attached end mass.The model considers the effect of elastic and translational motions of the beam.The elastic arm was assumed to reciprocate in a rigid prismatic joint. Al-Bedoor and Khulief[6]analyzed the dynamics of an elastic arm sliding in a rotating prismatic joint by using finite element technique.In thefinite element model all the inertia coupling terms were considered.Time depen-dent boundary conditions were used in order to account for the prismatic joint.The effect of an end mass was also taken into account in the model.Kalyoncu and Botsalı[7]solved a similar problem analytically by Lagrange formulation using the instantaneous eigenfunctions.Since the eigenfunctions and natural frequencies of an elastic robot arm were time dependent;the effect of the change of length of the robot arm on eigenfunc-tions and natural frequencies of the elastic robot arm were studied in this investigation.Yu¨ksel and Gu¨rgo¨ze [15]investigated theflexural vibrations of an axially moving robotic arm sliding through a prismatic joint while the joint was undergoing both vertical translation and rotary motion.Yang and Sadler[16]proposed a modal database procedure for analyzing the dynamics of a straight-line tracing manipulator.In most of the robot applications the robot hand carries a mass.This fact was not taken into consideration in Yang and Sadler’s investigation.Al-Bedoor and Khulief[17]formulated afinite element dynamic model for a sliding link through a prismatic joint where the prismatic joint was executing general planar motion.In contrast to previouslyM.Kalyoncu/Applied Mathematical Modelling32(2008)1087–10981089 reported formulations,afinite element mesh with afixed number of elements was used,where the element length was constant.Kalyoncu and Botsalı[18]investigated the effect of axial shortening on the bending vibra-tions of an elastic robot arm sliding in a rotating prismatic joint.Chalhoub and Chen[19]presented a general approach to systematically derive the equations of motion offlexible open kinematic chains.The methodology used the serial characteristic of the kinematic chain by complementing the4·4Denavit–Hartenberg transfor-mation matrix with a4·4structuralflexibility matrix.The latter was defined based on a rotating coordinate system which rendered the formulation applicable to both prismatic and revolute joints.Yu¨ksel and Gu¨rgo¨ze [20]investigated the vibrations of an axially movingflexible beam sliding through an arbitrarily driven pris-matic joint,restricted to move on a horizontal plane.Bauchau[21]focused on modelling of prismatic joints inflexible multi-body systems.In the classical formulation of prismatic joints for rigid bodies,kinematic con-straints are enforced between the kinematic variables of the two bodies.A sliding joint was proposed that involves kinematic constraints at the instantaneous point of contact between the sliding bodies.Kalyoncu and Botsalı[22]analyzed lateral and torsional vibrations of a robot manipulator with an elastic arm sliding in a prismatic joint.The mass of the end-effector was assumed as a point mass attached at the end of the elastic arm.Kalyoncu and Botsalı[23]analyzed lateral and torsional vibrations of elastic robot manipulators with prismatic joint.The elastic arm was assumed to carry a time varying end mass.In all of the investigations on dynamics of elastic beam with end mass,the end mass was assumed to be constant during the motion. However,in many engineering applications there exists a need for investigating the dynamics of an elastic beam sliding in moving prismatic joint with a time varying end mass.The distinctive aspect of this work was consid-eration of the effect of time varying end mass in dynamic analysis of beams sliding in a rotating prismatic joint. Ankaralıet al.[24]modeled a single linkflexible robotic arm which was driven by aflexible shaft.Theflexible arm was assumed to carry an end mass.Hamilton’s principle was used in obtaining the dynamic model.Basher [25]investigated the dynamic modeling of a single-linkflexible beam having both rotational and translation motions and the effects of higher-order dynamics on the response of the beam.An analytical model of the beam,characterized by an infinite number of modes,was developed using Euler–Bernoulli beam equation and modal expansion method.The infinite-dimensional transcendental transfer function for the manipulator was formulated without modal approximation that was conveniently transformed into state-space form.Farid and Salimi[26]proposed an inverse dynamic approach to determine the required actuating torque and force for a planarflexible-link manipulator with revolute-prismatic joints such that its end-point follows a given trajec-tory.The formulation includes all of the non-linear terms due to large rotation of the links.Since one of the joints is prismatic,the resulting system of equations is time varying too.Khadem and Pirmohammadi[27] considered a mathematical model capable of handling a three-dimensional(3-D)flexible-degree of freedom manipulator having both revolute and prismatic joints.This model was used to study the longitudinal,trans-versal,and torsional vibration characteristics of the robot manipulator and obtain the kinematic and dynamic equations of motion.The kinematic and dynamic equations of motion representing longitudinal,transversal, and torsional vibration characteristics solved in parametric form with no discretization.Stoenescu and Marghitu[28]investigated the effect of prismatic joint inertia on dynamics of planar kinematic chains with fric-tion.The mathematical model of a planar kinematic chain consisting of a prismatic joint sliding along a link that was connected to a revolute joint was developed using Lagrange’s equations.The influence of the slider inertia on the position of the application point of the joint forces was analyzed.The influence of the slider link inertia on the dynamic response of a spatial robot arm with feedback control was analyzed using Kane’s for-mulation.Akbaba and Yu¨ksel[29]investigated an elastic beam sliding through a prismatic joint having an arbitrarily given motion in afixed coordinate system.It was assumed that elastic beam was undergoingflexural in two planes and torsional elastic displacements,and elastic beam having a disk at one-end.Kinematics of prismatic joint was given as a3-D-space motion in afixed coordinate system.The prismatic joint was subjected to a propelling force from prismatic joint.With these assumptions,the differential equations of motion of the elastic beam were derived by using Hamilton’s Principle.Besides,the equations of motions of the system were derived as ordinary differential equations by using the assumed modes method.Most of the investigations on dynamics offlexible manipulators with revolute or prismatic joint considered the robot hand to follow a non-linear or non-straight-line path[30–33].Dynamics of straight-line tracing manipulators were not taken into consideration in most of the cited investigations.But,in some applications such as welding and painting,robotic arm has to follow predefined straight-line paths in the work volume.Inmany welding operations,the path followed by the robot hand during the process is a straight-line.In this type of operations,deviations from the predefined straight-line path may cause severe distortions in welded parts and also poor welding quality.Deviations from the predefined straight-line path can be considered as param-eters to predict welding quality in such operations.Furthermore,in cases where savings from energy and time are points of interest,the manipulator hand is expected to move through a linear trajectory between two points.As a result,dynamic analysis of flexible link manipulators with straight-line tracing hand is a signif-icant problem for robot design and control purposes.The aim of this study is investigation of dynamics of a flexible arm tracing multi-straight-line path.The flex-ible arm traces a multi-straight-line path under the action of an external driving torque and an axial force.Flex-ible arm is assumed to be an Euler–Bernoulli beam carrying an end-mass.Equations of motion of the flexible manipulator are obtained by using Lagrange’s equation of motion.In the formulation,the prismatic joint is treated as rigid.The flexible arm is housed inside the rigid prismatic joint.Effect of rotary inertia,axial short-ening and gravitation is considered in the analysis.Equations of motion are solved by using fourth order Runge–Kutta method.Numerical simulations are carried by using developed computer program.Physical trend of the numerical results are discussed in order to demonstrate the validity and the accuracy of the analysis.2.Dynamic modelThe physical configuration of the flexible robot manipulator considered in this study is given in Fig.1.The flexible arm is assumed as an Euler–Bernoulli beam.The mass and flexible properties are assumed to be dis-tributed uniformly along the flexible arm.The prismatic joint is assumed to be rigid.The flexible arm slides in the prismatic joint under the action of axial force F .The sliding motion of the flexible manipulator is assumed to be frictionless.The initial length of the beam is denoted as l 0and the variation in the length of the manip-ulator is denoted as u .Torque T rotates prismatic joint about Z axis.The flexible arm experiences a combi-nation of rotational and translational gross motion.Since,Lagrange’s equation of motion is to be used in obtaining the equations of motion;energy terms have to be evaluated.Homogeneous transformation matrices are used in the kinematic analysis.XYZ is the global reference frame,while x 0y 0z 0is the rotating and xyz is the rotating and translating reference frame.Angle between the rotating reference frame x 0y 0z 0and the global reference frame XYZ is denoted as h .In order to obtain the velocity terms;an infinitesimal mass dm on the flexible arm is considered as seen in Fig.1.Dis-tance of dm to the origin in x direction is u +x and the displacement from the undeformed position in y direc-tion is denoted as g .Then,coordinates of dm with respect to the global reference frame XYZ can be written as:X Y ¼cos h Àsin h sin h cos h u þxg:ð1Þ1090M.Kalyoncu /Applied Mathematical Modelling 32(2008)1087–1098Derivative of g with respect to time is written as:d g d t ¼o g o t þ_u o go x¼_gþ_u g 0:ð2ÞDifferentiating the expression in Eq.(1)and using Eq.(2)we obtain:_X ¼ð_uÀg _h Þcos h Àðu þx Þ_h þð_g þ_u g 0Þj ksin h ;_Y ¼ð_u Àg _h Þsin h þðu þx Þ_h þð_g þ_u g 0Þj kcos h By using obtained velocity terms,velocity of dm can be written as:V 2¼_X2þ_Y 2¼½ðu þx Þ2þg 2 _h 2þ_g 2þ_u 2þ2_g _u g 0þ_u 2g 02þ2_h ½ðu þx Þ_g þðu þx _u g 0Àg _u Þ :ð3ÞKinetic energy of the flexible manipulator is written as:XK ¼0:5Z l 0ÀuV 2dm þ0:5M e V 2l þ0:5J 0_h 2:ð4ÞPotential energy due to flexible deformations is written as:U d ¼0:5Zl 0ÀuEI o 2g o x 22d xð5Þand the gravitational potential energy is written as:U g ¼Z l 0ÀugYdm þgY l M e ð6Þwhere Y =(u +x )sin h +g cos h and Y l is the ordinate of position of M e at x =l 0.Components of the acceleration and the gravitational acceleration in x 0direction have to be considered in order to express the potential energy due to axial shortening.The absolute acceleration is:a ¼_Vþ_h ÂV ;ð7Þwith respect to the global reference frame.Where _his the angular velocity of the prismatic joint.The compo-nent of absolute acceleration in x 0direction can be written asa x ¼€u À€hg À€h 2ðu þx Þ:ð8ÞThe component of gravitational acceleration in x 0direction is expressed as:g x ¼Àg sin h :ð9ÞPotential energy due to axial shortening can be written by using the axial inertial force:F a ¼M e ðÀa x l þg x Þþq A Z l 0ÀuðÀa x þg x Þd xð10Þand Eqs.(8)and (9)as follows:U a ¼0:5Z l 0Àug 02F a d x :ð11ÞTotal potential energy of the flexible manipulator is obtained by summing the expressions in Eqs.(5),(6)and (11)leading toXU ¼U d þU g þU a ð12ÞM.Kalyoncu /Applied Mathematical Modelling 32(2008)1087–109810913.Equations of motionUsing the mode shapes of a clamped-free beam,flexible displacements of the manipulator can be written in the form of an infinite series:g ðx ;t Þ¼X 1i ¼1/i ðx ;t Þq i ðt Þ¼½/ f q gð13Þwhere {q };denotes the time dependent generalized coordinate,[/];denotes the time and space dependentassumed modes of the system [21]:/¼cosh kðx þu Þðl 0þu ÞÀcos k ðx þu Þðl 0þu ÞÀcosh k þcos k sinh k þsin k sinh k ðx þu Þðl 0þu ÞÀsin k ðx þu Þðl 0þu Þ:ð14ÞAssumed modes satisfy only the geometric boundary conditions of the system.Dynamic boundary condi-tions will be imposed by the kinetic and potential energy grange’s equation of motiond d t o L o _q Ào Lo q ¼F ð15Þis used in obtaining the equations of motion.Energy expressions are inserted in Lagrange’s equation ofmotion in order to obtain the equations of motion.By using the expressions for derivatives and squares of g given belowg 2¼f q g T½/ T½/ f q g_g2¼f q g T ½_/ T ½_/ f q g þf _q g T ½/ T ½/ f _q g þ2f _q g T ½/ T ½_/ f q g _g02¼f q g T _/0h i T ½_/0 f q g þf _q g T ½/0 T ½/0 f _q g þ2f _q g T ½/0 T ½_/0 f q g g 0¼½/0 f q g_g¼½_/ f q g þ½/ f _q g g 02¼f q g T ½/0 T ½/0 f q g _g0¼½_/0 f q g þ½/0 f _q g ð16Þa series of mathematical manipulations are performed to obtain the system equations as follows [18]:½M €q €u €h 8><>:9>=>;þ½H _q _u_h 8><>:9>=>;þ½G q u h 8><>:9>=>;¼0F ðt ÞT ðt Þ8><>:9>=>;;ð17Þwhere½M ¼f ðl 0;q ;A ;M e ;u ;½I ;½C i Þ;½H ¼f ðl 0;q ;A ;M e ;u ;_u;½I ;½C i Þ;½G ¼f ðl 0;q ;A ;E ;M e ;u ;_u ;€u ;½I ;½C i ;½_C i Þð18Þand [C i ]denote a square 1·1matrix [22].4.Mathematical modelling of the straight-line pathA flexible robot manipulator tracing a straight-line path through prescribed points P 0,P 1,...,P i is seen inFig.2.Tip of the flexible manipulator experiences P 0P 1,P 1P 2,...,P i À1P i ,...straight-line paths respectively.Joint angle h varies continuously during the motion.Value of h starts from 0and varies depending on the input function fed through a servomotor.In order to trace a linear path,length of the flexible manipulator1092M.Kalyoncu /Applied Mathematical Modelling 32(2008)1087–1098is changed depending on h and so by time.This fact necessitates the determination of variation of length of the flexible manipulator.From Fig.3angles h iÀ1and h i can be written as:h iÀ1¼arctanðY iÀ1=X iÀ1Þ;h i¼arctanðY i=X iÞ;ð19Þu iÀ1and u i are can be expressed as:u iÀ1¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX2iÀ1þY2iÀ1q;u i¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX2iþY2iqð20Þh iÀ1,h i,u iÀ1,and u i values at the interpoints are known since the followed path is prescribed.The distance P iÀ1P i can be written as:P iÀ1P i¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðX iÀX iÀ1Þ2þðY iÀY iÀ1Þ2q:ð21ÞAngle dOP iÀ1P i can be written as:dOP iÀ1P i¼arccosðP iÀ1P2þu2iÀ1Àu2iÞð2P iÀ1Pu iÀ1Þ:ð22ÞM.Kalyoncu/Applied Mathematical Modelling32(2008)1087–10981093Using Eqs.(19)–(22)_u and €u are obtained as:u ¼u i À1sin d OP i À1P isin 180Àd OP i À1P i Àh þh i À1Àl 0;ð23Þ_u ¼_hu tan 180Àd OP i À1P i Àh þh i À1;ð24Þ€u ¼€h _h _uþ_u2u þ_hh u sin 2180 Àd OP i À1P i Àh þh i À1:ð25Þ5.Numerical exampleA typical problem is solved for implementing the developed method.The tip of the flexible manipulator isassumed to trace straight-line paths defined by prescribed points P 0(1.8,0),P 1(2,2),P 2(0,3).The physical characteristics for the flexible manipulator are given in Table 1.h is assumed by the cycloidal functionh ¼h Tot t p t Àt p 2p sin 2p t p t ;ð26Þwhere h Tot is the total angle of motion,t p is the total time for motion,t is time.Total time for the motion t p isassumed as 15s and 20s.The tip mass M e is assumed as 2kg.First,two coupled modes has been considered in the analysis.The tip of the flexible manipulator is located at P 0at the beginning of the motion and moves to P 2along a multi-straight-line path,passing through the intermediate point P 1.Variation of u +l 0versus time in order to trace the prescribed path is determined and given in Fig.4.As seen from Fig.4,the curve for u +l 0consists of two piecewise continuous regions.Variation of the first three natural frequencies of the flexible arm versus time is given in Fig.5.As seen from Fig.5,natural frequency curves consist of two piecewise continuous regions as is case for variation of length of the flexible arm.All the natural frequencies decrease with increasing arm length as expected.Table 1Physical and geometric characteristics of flexible robot manipulator ParameterSymbol ValueUnit Densityq 2729.5kg/m 3Modulus of elasticity E 6.626·1010N/m 2Cross sectional area A 0.001471m 2Moment of inertiaI1.14197·10À08m41094M.Kalyoncu /Applied Mathematical Modelling 32(2008)1087–1098M.Kalyoncu/Applied Mathematical Modelling32(2008)1087–10981095Plot of tip displacement of theflexible manipulator is given in Fig.6.At t=0tip displacement isÀ5mm [6].Characteristic of tip deflection of system has two different regions.Since,change of u is dissimilar for P0P1 and P1P2paths,characteristics of the vibration of the elastic arm in P0P1path and P1P2path are distinct. Vibration amplitude in P1P2region is greater than the vibration amplitude in P0P1region.Sudden increase in the manipulator length increases the dynamic forces thus increasing the vibration amplitude.Vibration fre-quency in P1P2region is smaller than the vibration frequency in P0P1region.This is the expected case since the natural frequency of theflexible manipulator decreases by increasing manipulator length.As seen from Fig.7,decreasing total time of motion increases the vibration amplitude and decreases the vibration1096M.Kalyoncu/Applied Mathematical Modelling32(2008)1087–1098frequency.All numerical results given in Figs.4–7come out to be as expected physically.The same trend can be seen in results of Refs.[8–14,6,7,15–20,1,21,22].In Figs.8and9,the trajectory traced during the motion in P0P1and P1P2regions by the tip end of the flexible manipulator for the total motion time t p=20s and t p=15s are given respectively.6.ConclusionMathematical modelling and dynamic response of aflexible robot manipulator with rotating-prismatic joint are investigated in this study.The tip end of theflexible manipulator is assumed to trace straight-line paths passing through prescribed points.Theflexible robot manipulator is assumed to carry an end mass.Equations of motion of theflexible manipulator are obtained by using Lagrange’s equation of motion. Effect of rotary inertia,axial shortening and gravitation has been considered in the dynamic model.Equa-tions of motion are solved by using assumed modes and the principle of separation of variables.Equations of motion are solved by using fourth order Runge–Kutta method.Numerical simulations obtained by using a developed computer program are presented in graphic form and physical trends of the results are discussed.If the number of prescribed points is i,displacement of the tip end of theflexible manipulator is obtained for iÀ1piecewise continuous regions.Discontinuities occur just at the prescribed points.These discontinu-ities are observed as sudden variations in the length of theflexible manipulator and discontinuities in the nat-ural frequencies.Sudden changes in the manipulator length increase the dynamic forces and thus amplitude of vibrations and decrease frequency of vibrations as expected.。
