喜歡就充值下載吧。。資源目錄里展示的文件全都有，，請放心下載，，有疑問咨詢QQ:414951605或者1304139763 ======================== 喜歡就充值下載吧。。資源目錄里展示的文件全都有，，請放心下載，，有疑問咨詢QQ:414951605或者1304139763 ======================== Int. J. Dynam. Control (2015) 3:9499DOI 10.1007/s40435-014-0096-3Work cycle optimization problem of manipulatorwith revolute jointsBogdan Posiadala Mateusz Tomala Dawid Cekus Pawe Wary sReceived: 25 February 2014 / Revised: 27 March 2014 / Accepted: 4 April 2014 / Published online: 5 May 2014 The Author(s) 2014. This article is published with open access at SAbstractInthisworktheproblemofmotionmodelingandwork cycle optimization of manipulator with revolute jointshas been considered. The motion equations of the manipu-lator elements under any spatial work cycle conditions havebeen formulated. The formulation has been completed byusingtheclassicvectormechanicsandLagrangeequationsofsecondkind.Theequationsofmotionofthesystemhavebeenobtained using commercial software. The chosen motionmodel for each considered actuator is point-to-point motionmodelwithquasi-trapezoidvelocityprofile.Additionally,theproblem of optimization of a particular work cycle has beenpresented. The optimization objective has been chosen asminimization of loads (torques) in actuators. The objectivefunctionhasbeenformulatedusingperformanceindexesandthe design variables are rated velocity value and initial timevalue of work cycle in each considered actuator. The for-mulated optimization problem has been solved using con-strained Multi-Objective Particle Swarm Optimization algo-rithm. A numerical computation has been completed usingspecially performed software and results of the computationhave been attached to the paperwork.KeywordsModeling Dynamics Robotics Manipulator Motion Optimization1 IntroductionProblems of modeling and analysis of dynamical phenom-enon in multibody systems have been the subject of manyworks. In works 13, authors of this papers present theB. Posiadala M. Tomala (B) D. Cekus P. Wary sInstitute of Mechanics and Machine Design Foundations,Czestochowa University of Technology, Czestochowa, Polande-mail: tomalaimipkm.pcz.plproblem of modeling and analysis of dynamics of a truckcrane and its components. From the viewpoint of this work,itisworthtoquoteworks47.Intheworks,theproblemsofmodeling and optimization of robot manipulators have beenpresented with different objective functions and constraintsapplied to the algorithm.In this work, the problem of modeling of the dynamics of4R manipulator has been presented. Additionally, the prob-lemofoptimizationofthepoint-to-pointworkcyclehasbeenformulated and solved. An exemplary computation has beenperformedandresultsofthecomputationhavebeenattachedto the paperwork.2 Kinematics and dynamics of the manipulatorThe manipulator with four revolute joints (4R manipulator)allows positioning an end-effector of the manipulator in athree dimensional workspace and, additionally, allows rotat-ing the gripping device attached to the manipulator. The sys-tem of this kind is an open kinematic chain and has beenpresented in simple form on Fig. 1.The kinematics and dynamics of the considered systemhas been formulated in Cartesian global coordinate systemOXYZ, as shown on Fig. 1. The model of the manipulatorconsists of four rigid bodies connected by 1-DOF revolutejoints P, Q, S and N. All functions of kinematics have beendetermined using classical mechanics by introducing localcoordinate systems permanently attached to bodies of theconsidered kinematic chain. The problem of kinematics ofopen kinematic chains is widely described in works 812.The problem of inverse dynamics of robot manipulatorswith revolute joints involves determining the torque varia-tions in each considered joint, whereas the position, velocityand acceleration functions are known. The best way to solve123Work cycle optimization problem95Fig. 1 The scheme of the 4R manipulatorthe problem is to formulate proper functions of mechani-cal (kinetic and potential) energy and to use Lagrange equa-tions of second kind. If L is the Lagrangian, the equations ofdynamics of considered manipulator are:ddt?L qi?Lqi= Mi,i = 1.4(1)The generalized coordinates are:q = 1,2,3,4(2)TheLagrangianistotalkineticenergyofthesystemminusitstotalpotentialenergy.Becauseeachelementofthesystemis considered as a rigid body, a kinetic energy of a particularelement is a sum of kinetic energy in translational and rota-tional motions. A potential energy of a particular element issimply weight of the element multiplied by a distance to theminimum of potential energy (the OXY plane of the globalframe).In this work, the problem of optimization of 4R manipu-lator is also presented. The optimization objective is to mini-mizetorquesineachconsideredactuator.Theobjectivefunc-tion can be formulated using a performance index 12. Fora particular actuator, the index has a form:Pi=tk?0M2idt(3)3 Motion modelIn this work, the point-to-point model of motion has beenaccepted. In literature, various models of velocity profile canbe met. For instance, the profile can be chosen as trapezoid,sinusoidal or parabolic 12. In this work a quasi-trapezoidFig. 2 Angular velocity versus time for chosen motion modelFig. 3 Angular acceleration versus time for chosen motion modelvelocity profile has been taken. Velocity and accelerationtime variations are shown on Figs. 2 and 3. On the figures,all important work cycle parameters are presented.From the standpoint of optimizing, the most importantparameters are starting time of the work cycle and its ratedvelocity. An angular displacement in each considered jointcan be simply computed as:si= (B)i (A)i(4)A maximum acceleration and duration of rated velocitykeeping are equal to:ai=vit(z)i+ t(a)i(5)t(d)i=?sivi? t(z)i 2t(a)i(6)Design variables can be collected to a vector:x = v1,v2,v3,v4,t1,t2,t3,t4(7)4 Particle swarm optimization algorithmTheparticleswarmoptimizationalgorithmisoneofthemostmodern stochastic optimization technique, it was firstly pro-posed by Kennedy and Eberhart in 1995 in work 13. Fromthe beginning, this method is widely developed and manyapplications and modifications have been formulated so far,forinstance1416.Inrobotics,thismethodisoftenusedtofind optimal geometrical and inertia parameters of stationaryrobots, such as manipulators 47. It is also used in mobileroboticstofindanoptimaltrajectoryformobilerobotintwo-dimensional workspace.The PSO algorithm is based on observation of phenom-enon occurring in nature, such as foraging of swarm ofinsects or shoal of fish. Each particle of the swarm is ableto remember and use its experience that is taken from the12396B. Posiadala et al.Fig. 4 A simplified scheme ofthe constrained PSO algorithmwhole iteration process and, also, is able to communicatewith other members. The swarm of particles is able to iden-tify “good” areas of the domain and can search deeply inthese areas for an optimum.Initialvaluesofdesignvariables(thepositionofparticularparticle)arerandom.Then,inaniterationstepn+1,distancecovered by a particle in m-th direction (the velocity of theparticle in m-th direction) can be described as following:V(n+1)m=?wV(n)m+ c1r1?pm x(n)m?+ c2r2?gmx(n)m?,(8)where isaconstrictionfactor, V(n)misavelocityinpreviousiteration step, w is a weight coefficient,r1andr2are randomreal numbers taken from (0;1), c1and c2are learning fac-tors, pmis a personal best position of the considered particlefromthewholeiterationprocessand gmisaglobalbestposi-tion obtained by entire swarm. In the formula, three differentinfluence components can be identified: the first is an inertiainfluence, the second is a personal influence and the third isa social influence. There is another version of this formula,where the global best position gmis replaced with a localbest position lm. In this version, each particle has specifiedneighborhood and compares its personal best position withmembers of the neighborhood only.Additionally, a maximum velocity in each considereddirection should be set to protect the swarm from explosion:V(n+1)m=V(n+1)msgn?V(n+1)m?V(max)mif?V(n+1)m? V(max)m,(9)where V(max)mis the maximum velocity in m-th direction.A new position for each particle in each considered direc-tion is equal to:x(n+1)m= x(n)m+ V(n+1)m.(10)During the iteration, values of design variables have tosatisfy some constraints. All variables have to be positive.Signs of the velocities are known and depend on signs of theangular displacements in each considered actuator (Chapter3). Additionally, the velocities are constrained by maximumvelocities thatareavailable ineach actuator.Moreover, max-imum time of work cycle is specified and a maximum torquevalue is known for each joint. All formulated constraints arelisted below:0 ti t(max)i(11)0 vi 0(13)|Mi| M(max)i(14)Considering previously introduced model of motion(Chapter 3):t(max)i= tk t(d)i 2t(z)i 4t(a)i(15)In PSO algorithm, constraints are introduced using apenalty function. There is necessary to reformulate con-straints into a following form: (x) 0(16)The penalty function can be assumed as:F (x) = h (n)?r?k=1pkmax (k(x)?(17)123Work cycle optimization problem97wherer isnumberofconstraints.InEq.(17), isacorrectingfactor for the penalty function and pkare correcting factorsfor each considered constraint. The max function takes thevalueof0whenk(x)0.Theh(n)function depends on the iteration step and has been assumedas:h (n) = nn(18)Using the considerations contained in Sections 2 and 3, theobjective function for unconstrained optimization problemcan be formulate as:f (x) = min4?i=1wiPi(x),(19)4?i=1wi= 1,(20)where wiare weight coefficients and Pifunctions aredescribed by Eq. (3). After introducing the penalty functioninto the objective function, a new objective function is givenby a following formula:f (x) = min?4?i=1wiPi(x) + F (x)?(21)The PSO algorithm, used in this work, has been presentedin a simplified form in Fig. 4.5 The exemplary computationThe algorithm, introduced in previous chapters, has beenused to perform the exemplary computation. The problem ofoptimization of the work cycle of 4R manipulator has beeninvestigated. The design variables are starting time and ratedvelocity in each considered joint, so there are 8 design vari-ables. The work cycle includes a motion from a start point Ato a final point B with simultaneous rotation of the grippingdevice and a load. The gripping device and the load are con-sidered as one rigid body. Cartesian coordinates of chosenpointsareA0.5,0.2,0.8andB1.3,0.9,0.5.Theinversekinematics problem for the 3R manipulator has been solvedand specific rotation of gripping device has been set from/3 to /3 rad. Complete inverse kinematics results are:qA= 0.38051,0.63141,2.54841,1.04720(22)qB= 2.53605,0.64876,1.54163,1.04720(23)There is assumed that the center of mass of particular bodyis placed in the half of the bodys length. Geometrical andinertiaparametersare:l1=0.4m,l2=1m,l3=0.8m,l4= 0.4 m, m1= 0.7 kg, m2= 1.1 kg, m3= 0.8 kg,m4=1.5 kg, J1= 0.05,0,0,0,0.03,0,0,0,0.05 kg m2,J2= 0.1,0,0,0,0.1,0,0,0,0.002 kg m2, J3=0.1,0,0,0,Fig. 5 Torque variations versus time for the solution of optimizationcase 1Fig. 6 Angular velocity variations versus time for the solution ofoptimization case 10.1,0,0,0,0.002 kg m2 J4=0.05,0,0,0,0.05,0,0,0,0.001kg m2. Parameters of motion model are: t(a)i=0.05 s,t(z)i= 0.1 s, tk= 10 s. Maximum angular velocitiesand torques has been assumed as: v(max)i=1.15 rad/s,M(max)1= 30 N m, M(max)2= 80 N m, M(max)3= 40N m, M(max)4= 20 N m. The chosen PSO parametersare: = 0.8, c1=2.1, c2= 2.0, w =0.6, = 30, pk=1,V(max)1= V(max)2= V(max)3= V(max)4= 0.1 rad/s,V(max)5= V(max)6= V(max)7= V(max)8= 0.5 rad/s. Num-ber of particles is set to 500 with 100 iterations in each case.The integration step has been set to 0.005 s.Four different cases of optimization have been investi-gated for different values of wages wi. In the first caseall the weight coefficients are equal: wi= 0.25. For thiscase, the final value of objective function is 672.63 withindividual values of performance indexes: P1= 3.52, P2=1487.58, P3= 1201.36, P4= 11.4 106. The designvariables are 0.90,0.13,1.15,0.67,3.37,0.00,8.92,3.87. Inthe second case the weight coefficients are: w1= 0.1, w2= 0.5, w3= 0.3, w4= 0.1 and the results of optimizationare 1257.52 and 0.38,0.84,0.78,0.87,0.90,8.27,0.00,4.75.For this case, individual performance indexes are: P1=0.36, P2= 1170.88, P3= 2236.31, P4= 9.46 106.In the third case the weight coefficients have been cho-sen as: w1= 0.1, w2= 0.3, w3= 0.5, w4= 0.1. Theobtained result of objective function is 945.61 with indi-vidual values of performance indexes: P1= 6.83, P2=2794.95, P3= 213.173, P4= 14.4 106and designvariables 0.33,1.06,1.15,1.06,3.06,0.00,8.92,3.60. In thefourth case the weight coefficients are: w1= 0.05, w2=0.55,w3=0.35,w4=0.05andtheresultsofoptimizationare1236.84 and 0.69,0.13,1,15,0.78,2.34,0.02,8.92,2.41. Forthis case, individual performance indexes are: P1= 1.92, P2= 1475.86, P3= 1214.36, P4= 7.89106. All the resultshave been presented as torque time-variations for each con-12398B. Posiadala et al.Fig. 7 Torque variations versus time for the solution of optimizationcase 2Fig. 8 Angular velocity variations versus time for the solution ofoptimization case 2Fig. 9 Torque variations versus time for the solution of optimizationcase 3Fig. 10 Angular velocity variations versus time for the solution ofoptimization case 3Fig. 11 Torque variations versus time for the solution of optimizationcase 4Fig. 12 Angular velocity variations versus time for the solution ofoptimization case 4sidered case (Figs. 5, 7, 9, 11). Additionally, velocity time-dependencies has been added (Figs. 6, 8, 10, 12).6 ConclusionInthiswork,theproblemofmodelingandoptimizationofthedynamics of 4R manipulator has been presented. The point-to-pointmotionmodelwithaquasi-trapezoidvelocityprofilehas been accepted. Equations of motion have been obtainedusing classical vector mechanics and Lagrange equationsof second kind. The optimization problem has been solvedusing constrained Multi-objective particle swarm optimiza-tionalgorithm.Thedesignvariablesareratedvelocityineachactuator and starting time of the work cycle. The objectivefunction is based on the minimization of torques in actuatorsusing performance index.The algorithm can be used to investigate other optimiza-tion problems with different objective function and differentdesign variables. The consideration can be used for solv-ingoptimization problem formanipulators withrevolute andprismatic joints, the key is to reformulate the objective func-tion and to identify design variables.AcknowledgmentsThe study has been carried out within statutoryresearch BS/PB-1-101-3010/13/P of the Institute of Mechanics andMachine Design Foundations of the Cze stochowa University of Tech-nology. 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