Contributions to the cinematic and dynamic study of parallel mechanism robots with four degrees of freedom

Transcription

1 Equation Chapter Setion Forgó Zoltán, eng. hd. andidate Contributions to the inemati and dynami study of parallel mehanism robots with four degrees of freedom Abstrat of hd hesis Sientifi oordinator: Niolae litea, prof. hd. eng. 28

2 Abstrat At the very beginning the men tried to make his life more omfortable, but in the same time the quality of living should be inreased. his dream was true at the seond part of the 2 th entury by the massive use of robot tehnology. he new era introdued this onept in the industry and in the all day life as well. From point of view of kinemati struture two type of robots are used in the prodution tehnology: the open and the losed loop mehanisms. his seond one is known as the parallel robots. As Merlet wrote in many artiles [MER97], [MER], [MER2], [MER4] the dimensional and topologial synthesis of this mehanisms is an important area of siene to be onsidered. oday about hundred topologies are known and studied but this is not the whole possibility, there are more kinemati and dimensional onfigurations with must be investigated. Some methods for this job are known as the Gruëbner mobility theory (mentioned by Merlet [MER], sai [SA99], the group theory (Hervé [HER78], Martinez [MAR3], Rio [RIC2], the enumerative method (Chakarov [CHA98], Rao and others. o be able analyzing and generating suh mehanisms the use of Lie group theory (Hervé [HER99], the srew theory (-Kong [KON], [KON6], Fang [FAN2], Carriato [CAR5], exponential matries method (extended by Negrean [NEG2], [NEG4]. [NEG5], the linear transformation method (Gogu [GOG5], [GOG6] and others are used. From mobility point of view three groups of four degrees of freedom mehanisms are known: with three rotations about the axis and one translation about one axes, one rotation and three translations respetively two and two mobility around and about two axis. his last one doesn t have important industrial appliation so it is more less investigated. he present thesis introdues a novel 4 dof arhiteture for the seond group (the so alled Shoenflies motion and the respetive analysis is done involving analytial alulations, original mathematial and simulation algorithms and Matlab Simulink model solving as well. After introdution in the state of the art of parallel robot analysis the original mathematial method is presented: geometrial and kinematial investigation using matrix form losing equations. o define this matrix equation let onsider the Fig. A. he relative position and orientation of the {} and {ji} systems define the homogen transformation matries for the (i and (j body joint for rotation, translation, universal and spherial joints: os(q sin(q sin(q R os(q R (q R = ji z ji ( os( γzj sin( γzj (q zj zj sin( γ os( γ = ji q (2 sin(q + q os(q + q U U2 U U2 (q U,q U2 = ji os(qu + q U2 sin(qu + q U2 (3

3 ( ( ji {j} z j z G p ji z ji x ji j p j x j G j y j (j i p i x G ji z i G i {i} y i x i p j (i p i z o {} O y o α α α = jixx jiyx jizx βjixy βjiyy βjizy S S2 S3 ji γjixz γjiyz γjizz (q,q,q α = os(q os(q γ = + α = β = γ = jixx S2 S3 β jixy = sin(q Ssin(q S2os(q S3 + os(q Ssin(q S3 γ jixz = os(q Ssin(q S2os(q S3 + sin( αx sin(q S3 α jiyx = os(q S2sin(q S3 β jiyy = sin(q Ssin(q S2sin(q S3 + os(q Sos(qS3 jiyz os(q Ssin(q S2sin(q S3 sin(q Sos(q S3 jizx sin(q S2 jizy sin(q Sos(q S2 jizz os(q Sos(q S2 (4Next to this step a matrix equation an be written to determine the positionorientation of (j body relative to the {} system using homogen transformation matries: ( [ ] [ ] [ ] [ ] [ ] [ ] [ ] i j i j i ji j j = = (4 In ase of one kinemati hain the losure equation will be as follows: (q i ii+ (q i+ = and (5 x o Fig. A Relative position and orientation of body s to define matrix losure equation [ ] [ ] [ ] [ ] [ ] i+ i+ i+ + i n i+ i+ 2 ii+ + + i i= { } ( q,q,,q = [ ] [ (q ] [ ] (6 Considering a parallel robot with mobile platform having m degrees of freedom the homogen transformation matrix desribing the position-orientation in respet to the {} fixed system is in funtion of operational parameters αx, βy, γ z,x,y,z. Defining the following equations the diret and inverse geometry of the robot an be alulated:

4 ( 2 i n = ( αx βy γz q,q,,q,,q,,,x,y,z k ( q k,q k2,,q ki,,qkn = ( α k x, βy, γz,x,y,z k ( q m,q m2,,q mi,,qmn = ( α k x, βy, γz,x,y,z m he veloity and aeleration alulations an be obtained by deriving the equation (5 in respet to time. Using those results the unknown variables are alulated by the following equation: ( = ( = + ( = ( xq,q,i x q,q,q i vq,q,i ( = yq,q,i n a( q,q,q i y = (8 ( q,q,q i = (9 zq,q,i z( q,q,q i Rq ( i i,i= { ω( q i i,i= } = ( Rq ( i i,i= Rq ( i i Rq ( n i i,q i i + + { ε( q i i,q i i } = ( + R ( q i i R ( q n i i,q + i i he seond method to analyze the mehanisms is to use the CAD systems. he algorithm presented is implemented in Autdesk Inventor software and so it is possible to determine the joint displaement, veloity, aelerations and the kineti energy as well. he results are presented graphially (Fig. A2. and an be exported in Mirosoft Exel format. (7 Fig. A2 CAD analysis result graph and table

5 z B C B 2 A A 2 C 2 B 4 A 4 C 4 B 3 A 3 x C 3 he third hapter of the thesis presents the analysis and modeling of the proposed four degrees of freedom parallel robot. As the first step the topology of the parallel mehanism is defined (Fig. A3 and the workspae and singular onfigurations are investigated. Further the analytial equation for the geometri, kinemati and dynami modeling are established. he diret and the inverse problemati are onsidered. he inverse geometrial equations are presented as follows: 2L y + 4L (x + y (x + y q = 2 atan (2 2L x + x + y q Fig. A3 he kinemati sheme of the four degrees of freedom parallel mehanism q 2L y 4L (x + y (x + y 2 = 2 atan 2L x + x + y 2Ly ( BA + Ls θ ( + θ + ( + θ + ( + θ 2LxBA L xba L yba Ls 3= 2 Atan 2 2 4L ( xba + Lθ + ( yba + Lsθ ( xba + Lθ + ( yba + Lsθ 2Lx ( BA + L θ + ( xba + L θ + ( yba + Ls θ (3 (4 2π q4 = z +θ (5 ps where p s is the pith of the srew B. In the ase of dynami modeling using the Lagrange differential equation are used and firs the so alled fitive generalized fore vetor F fg is alulated. Using the Jaobian matrix the following equation is used to determine the generalized fore vetor for the ative axis of the parallel robot: fg = fg (6 Q J(q F

6 Calulated differene (deg.e+ -5.E-3 -.E-2-2.E-2-2.E-2-3.E-2-3.E ime (s Calulated differene (deg/s and (deg/se^2 8.E-3 6.E-3 4.E-3 2.E-3.E+ -2.E-3-4.E-3-6.E-3-8.E ime (s a Calulated differene (deg -5.E-2-6.E-2-6.E-2-7.E-2-7.E-2-8.E ime (s Calulated differees (deg/s and (deg/s^2 2.E-2.E-2 5.E-3.E+ -5.E-3 -.E ime (s b Fig. A4 Calulation differenes of joint parameters for joint A and A2 o prove the obtained methods a numerial example is imposed and the above desribed two analysis methods are onsidered as well. he obtained results are ompared and the differenes are presented in Fig. A4. From dynami point of view a Matlab-Simulink model was build and the same motions are imposed to obtain the dynami simulation. In this way the analytial autions are ompared to the Simulink model results. he differenes are presented in Fig. A5..5 Ative Dif. forta joint gen. A Ative Dif. forta joint gen. A2 2 Ative Dif. forta joint gen. A3 3 Ative Dif. forta joint gen. A t (s Fig. A5 he alulated differenes for the obtained dynami results

7 In the final paragraphs the physial model of the four degrees of freedom parallel robot is presented: the mehanial parts of the robot are introdued, the 24 DC atuators and the afferent eletrial parts are showed. he ommuniation with the personal omputer is realized via RS232 serial port. Using the interfae it is possible to move the robot using diret and inverse geometry input data. Importing a *.X file ontaining different onfigurations a path an be followed by the end-effetor of the robot. 4 dof mehanism atuators 24V DC ommand system Fig. A6 he omponents of the 4 dof parallel robot omuniation module (master SS SS2 ommand module (slave SS3 SS4 Current soure 5V, 24V - 6A D.C. Fig. A7 he omponents of the ontrol system of the 4 dof parallel robot