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Received December 20，2009；revised May 10，2010；accepted July 8，2010；published electronically October 20，2010Abstract：Currently，virtual assembly technology has attracted increasing aRention due to considerations of solving assembly problemsin virtual environment before actual assembly in manufactory．Previous studies on kinematic analysis of mechanism only aim atanalyzing motion law of single mechanism。but can not simulate the multi．mechanisms motion process at the same time．1et alonesimulating the automatic assembly process of products in a whole assembly workshop．In order to simulate the assembly process ofproducts in an assembly workshop and provide effective data for analyzing mechanical performance after finishing assembly simulationin virtual environment，this study investigates the kinematics analysis of mechanisms based on virtual assembly．Firstly．in view of thesame function of the kinematic pairs and the assembly constraints on restricting the motion of components(subassembly or part)，themethod of identifying kinematic pairs automatically based on assembly constraints is presented．The information of kinematic pairs canbe obtained through calculating the constraint degree of the assembly constraints．50296
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Secondly，the incidence matrix eliminating elementmethod is proposed in order to search the information and establish the models of mechanisms automatically after finishing assemblysimulation in virtual environment．Both methods have important significance for reducing the workload of pretreatment and promotingthe level of automation of kinematics analysis．Finally，the method of kinematics analysis of mechanisms is presented．Based onDescartes coordinates，three types of kinematics equations are formed．The parameters，like displacement，velocity，and acceleration，can be obtained by solving these equations．All these data are important to analyze mechanical performance．All the methods areimplemented and validated in the prototype system virtual assembly process planning(VAPP)．The mechanism models are establishedand simulated in the VAPP system．and the result curves are shown accurately．The proposed kinematics analysis of mechanisms basedon virtual assembly provides an effective method for simulating product assembly process automatically and analyzing mechanicalperformance after finishing assembly simulation．Key words：virtual assembly，assembly constraint，constraint degree，eliminating element method，kinematics equations，
1 IntroductionVirtual assembly technology is widely used presently inthe domain of machinery．It can guide the workers toassemble the products and solve the problems in virtualenvironment before actual assembly during manufacturingso as to improve the assembly quality ofproducts．Kinematics analysis of mechanisms based on virtualassembly improves the simulation abilities of virtualassembly system effectively,because it provides aneffective method for simulating multi．mechanisms motionprocess at the same time and important kinematic valuesafter finishing assembly simulation．However there are stillfew virtual assembly systems that have the function ofkinematics analysis of mechanisms．Hence．the method ofkinematics analysis of mechanisms based on virtualassembly draws wide attention at present．As all integrated technology,kinematics analysis of mechanisms based onvirtual assembly needs support of many technologies．Firstly,assembly constraints can provide contributions forestablishing kinematic pairs information automatically in acertain extent，because there is a common ground betweenthem on restricting motion of component(subassembly andpart)．In 1 992，TURNER，et al【l J，presented a unif01TIIrepresentation for assembly constraints that applies to awide variety of constraint relations．and also gave asystematic procedure for constraint reduction．In 1 997．RAJAN，et al【“，presented assembly representations tocapture the ioint kinematics information and componentdegrees of freedom(DOF)．In 2007，YANG，et al■presented the relation between assembly constraint andDOF．and defined the movable DOF in virtual environment．These studies provide important contributions for assemblyconstraint study in virtual environment． Secondly，establishing models of mechanisms automatically afterfinishing assembly simulation in virtual environment is alsovery important，because it reduces the pretreatmentworkload of kinematics analysis．Kinematics analysis of mechanisms，as another important technology，is a strongemphasis which has been studied for many years[4。1 11．In2003，HERNANDEZ，et al[io】，proposed a geometricalapproach to obtain a velocity equation valid for planar andspatial linkages，and provided a new method for formingthe velocity equation．In this paper，three important methods of kinematicsanalysis of mechanisms based on virtual assembly arepresented．(1)The method of identifying kinematic pairsbased on assembly constraints．It is very effective to reduceworkload of kinematics analysis pretreatment．(2)Themethod of establishing the models of mechanismsautomatically．It can provide necessary information formotion simulation and kinematics analysis of mechanisms．(3)The universal method of mechanisms kinematicsanalysis．Based on Descartes coordinates，three types of thekinematics equations are formed and all the necessarykinematics parameters such as displacement，velocity andacceleration can be obtained by solving these equations．These data can provide important value reference foranalyzing the kinematics characteristic of mechanisms．Kinematics analysis of mechanisms based on virtualassembly is an integrated technology which combines thetechnologies of virtual assembly and mechanismskinematics analysis together in one system．2 Assembly Constraint and Kinematic Pairwsdppp一oroeaKinematic pair is the necessary element of a mechanism．There is a common ground between assembly constraintand kinematic pair on restricting motion of component，sosome information of lower kinematic pairs can be obtainedfrom constraint information．Notion of constraint degreeand some parameters are defined as follows．Constraint degree D(ti，，，)is the degree of restrictingmotion of component by assembly constraint．D(ti，吩)contains translational constraint degree signed by☆=(ffl’to，矗3)，(％·％=0，P，q=l，2，3)(translation about i directionswhich arc in plumb is permitted)．and rotational cons打aintdegree signed by吩=(吩l，伽，rj3)，(饰。Fjq：0，P，q=l，2，3)(rotation about{axes or{directions which are in plumb ispermitted)．Four types of ti are distinguished in nlble 1 andthree types of r／are distinguished in Thble 2．Constraintdegree is not equal to the degree of freedom,becausedegree of freedom belongs to a model，but constraintdegree belongs to constraints． F is the operation of reduction of constraint degree．which contains reduction of translational constraint degreesigned by F(ti，tj)，and reduction of rotational constraintdegree signed by只％吩)．Rotation is permitted about any axis passing through thepoint Pdi,if Pdi=矽，the point does not exist．Rotation ispermitted about one axis passing through the point p一；ifp一=矿，the point does not exist．The constraint degree is a result of restricting the motionof a component which the constraints act on．The finalresult of the constraint degree can be obtained through theoperation ofreduction of cons仃aint degree．The detailed rules of calculation are as follows．(1)If there is only one constraint。the rule of calculation isexpressed as 3 Establishing the Models of MechanismsThere iS not any model of mechanism after finishingassembly simulation in virtual environment，because thefixed links have not been defined．Although the informationof links andkinematic pairs exists，it does not belong to anymechanism．
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