Open Access A Computer Aided Kineto-Static Analysis Method for 3-RRRT Parallel Manipulator Based on Vector Bond Graph

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1 Send Orders for Reprints to The Open Automation and Control Systems Journal,, 5, - Open Access A Computer Aided Kineto-Static Analysis Method for -RRRT Parallel Manipulator Based on Vector Bond Graph Zhongshuang Wang * and Yangyang Tao School of Mechatronics Engineering, Qiqihar University, Qiqihar, Heilongjiang 66, China Astract: For improving the reliaility and efficiency of the kineto-static analysis of complex root systems, the corresponding vector ond graph procedure is proposed. By the kinematic constraint condition, the vector ond graph model of universal are made. Based on this, the vector ond model of -RRRT parallel Manipulator is uilt. For solving the algeraic difficulties rought y differential causality in system automatic kineto-static analysis, the effective ond graph augment method is proposed. From the algeraic relations of input and output vectors in the asic fields, junction structure and Euler-junction structure of system vector ond graph model, the unified formulae of driving moment (or force) and constraint forces at s are derived, which are easily derived on a computer in a complete form. As a result, the automatic kineto-static analysis of -RRRT parallel Manipulator is realized, the validity of the method presented here is illustrated. Keywords: -RRRT Parallel Manipulator, Kineto-Static Analysis, Vector Bond Graph, Causality, Bond Graph Augment, Constraint Joint.. INTRODUCTION Parallel manipulators have een widely used in industry, such as machine tool, simulator for aircraft driver, automatic assemling for automoile production, and micromechanism for computer assisted surgery [, ], -RRRT parallel manipulator is one of important type of such systems. The kineto-static analysis is very important for the design, control, static and dynamic strength check of such system. Due to high nonlinearities and couplings involved in parallel manipulators, determining driving moment (or force) and the constraint forces at s is a very tedious and erroprone task. The Newton-Euler technique and Lagrange technique are two of the well known methods used for the dynamic analysis of a root system [, ]. These techniques however, are only suitale for a single energy domain systems, e.g. mechanical systems, and can not e used to tackle systems that simultaneously include various physical domains in a unified manner. The ond graph technique developed since the 96 s has potential applications in analyzing such complex systems and has een used successfully in many areas [, 4]. It is a pictorial representation of the dynamics of the system and clearly depicts the interaction etween elements, it can also model multi-energy domains, for example, the actuator systems, which may e electrical, electro-magnetic, pneumatic, hydraulic or mechanical. Once the ond graph model of the system is ready, the system dynamic equations can e derived from it algorithmically in a systematic manner. This *Address correspondence to this author at the School of Mechatronics Engineering, Qiqihar University, Qiqihar, Heilongjiang 66, China; wzhsh96@6.com process is usually automated y using appropriated software [, 4]. But for spatial multiody systems such as parallel manipultors with different constraint s, the kinematic and geometric constraints etween odies result in differential causality loop, and the nonlinear velocity relationship etween the mass center and point on a ody leads to the nonlinear junction structure. The traditional ond graph procedures mentioned aove were found to e very difficult algeraically in automatic modeling and kineto-static analysis of systems on a computer. To solve this prolem, the Lagrange multiplier approach and Karnopp-Margolis approach [5, 6] can e employed to develop ond graph model for multiody systems. Based on the Karnopp-Margolis approach, Zeid proposed a new singularly pertured explicit modeling method [7] and provided a more realistic ond graph model for revolute, prismatic, spherical, universal, and so forth ased on scalar ond graph concept [8]. The ond graph technique and causality concept have een well developed for systems in which components have scalar constitutive laws. But for spatial multiody systems, the scalar ond graph technique is found to e complex, difficult and requires a great amount of experience, ecause three dimensional motion of system components must e resolved into scalar onds. To address this prolem, the vector ond graph techniques [9-] were proposed. In vector ond graphs, single power onds are replaced y multipower onds, this makes it posses more concise presentation manner and e more suitale for modelling spatial multiody systems. Some prolems however, should e studied further, such as modeling complex spatial root mechanism with different constraint s y vector ond graphs, augmenting the vector ond model to avoid differential causality, developing the generic algorithm for automatic kineto-static / Bentham Open

2 4 The Open Automation and Control Systems Journal,, Volume 5 Wang and Tao Fig. (). The diagram of spatial universal. analysis of spatial root mechanism. To solve these prolems, a more efficient and practical computer aided kinetostatic analysis procedure for complex spatial root mechanism ased on vector ond graph is proposed here.. THE VECTOR BOND GRAPH MODEL OF UNI- VERSAL JOINT The diagram of spatial universal is shown in Fig. (). This allows only two direction rotations etween its joined ody B! and B!, fixing the remaining three translational and one rotational degrees of freedom. Therefore, only two generalized coordinates are free to change. Joint point P and Q are fixed on rigid ody B! and B! respectively, vector h is used to descrie the relative motion of the two rigid odies, h = r! P r # Q. Where r! P and r! Q represent the position vector of point P and Q in gloal coordinates respectively, r P = [ x P y P z P ], r x T y Q T z ] á á á á Q = [ Q. d! and d! are two unit vectors fixed on rigid ody B á and B!, which are parallel to axis, and orthogonal to each other. d! and d! are the corresponding vectors in ody frame. From the kinematic constraint condition of universal [], the position constraint equations can e written as! (u) (h,d,d # ) = % ( P Q r r # * * & d T # d )* = = () r P Q &! r # ) & ) d T # A #T A! %& d! () Q where A α and A β are the direction cosin matrices of ody B! and ody B! respectively. The corresponding velocity (angular velocity) constraint equations can e written as &! (u) (h, d, d # ) = where d n = A n d n d n = A n d n! á and! represent the angular velocity vectors of the rigid ody B determined in gloal coordinates,! and () á B á and! are the corresponding angular velocity vectors of the rigid ody determined in ody frame. %d! = %, d %! = % -d!z d! -d y!z d! y d!z -d!x d!z -d!x # -d! y d!x & # -d! y d!x & The velocity and angular velocity constraint equations shown in Eq. () can e presented y vector ond model shown in Fig. (), where the modulus matrices of MTF can e otained from Eq.() directly.. THE VECTOR BOND MODEL OF -RRRT PAR- ALLEL MANIPULATOR As shown schematically in Fig. (), a -RRRT parallel manipulator consists of a movale upper platform (Δ, a fixed ase platform (Δ and three parallel moving chains (A-a, B-, and C-c). Each chain consists of three rigid

3 A Computer Aided Kineto-Static Analysis Method The Open Automation and Control Systems Journal,, Volume 5 5 MTF MTF MTF MTF Ø Â Ø Á Ø Â Ø Á r Q Â P r Á Fig. (). The vector ond graph model of spatial universal. Upper platform c a Body Universal Body Body Body T A Body Body Body C Base platform T C Body Body A B T B Fig. (). The diagram of -RRRT manipulator. odies and is connected to the ase and upper platform y a universal and a revolute respectively, and three odies in a single moving chain are joined each other y two revolute s. The joined structure of this system is shown in Fig. (4). For chain A-a, B- and C-c, the corresponding driving moments are T A T B and T C respectively. Universal Body Upper platform Universal Base platform Universal Body Body Body Body Body Body Body Body Fig. (4). The joined structure of -RRRT manipulator. Fig. (5) shows the configuration of -RRRT manipulator which only contains a moving chain A-a for simplification. Gloal coordinates C p X Y Z is located at the equilateral triangle center Cp of ase platform, and ody frame C up X p Y p Z p is located at the equilateral triangle center C up of upper platform. The other ody frames is located at the mass centers of odies in three moving chains( A-a, B-, C-c ). For example, C ai X ai Y ai Z ai (i=,, ) represents the ody frame corresponding to ody i in chain A-a. The circumradius of upper platform is r, and R presents the circumradius of ase platform. In Fig. (5), è and ji è & ji represent relative angular displacement and relative angular velocity of ody i in chain j respectively ( i=,,, j=,, ). If the motion of upper platform center x up = f (t), y up = f (t) and z up = f (t) are given, è and ji è & ji can e determined y the kinematic analysis of mechanism []. Oviously, This manipulator has three degrees of freedom. From the kinematic relations of mass center velocity vector, velocity vector in gloal coordinates and angular velocity vector in ody frame, a spatial moving rigid ody can e modeled y vector ond graph []. The Newton- Euler equation descriing the rotation of a rigid ody contains a term with an exterior product etween angular momentum vector and angular velocity vector, this term can e expressed y vector ong graph and e called Euler-junction

4 6 The Open Automation and Control Systems Journal,, Volume 5 Wang and Tao Fig. (5). The configuration of -RRRT manipulator. Fig. (6). The vector ond model of moving chain A-a, B-, and C-c in -RRRT manipulator. structure (EJS) [, ]. For a spatial moving rigid ody i in chain j ( i=,,, j=,, ), the relationship etween input vector and output vector of its EJS can e written as E out ji = (! )! ji = (! )E in () ji where E out = [E ji out E jix out E jiy out ] T, E jiz in =! ji = [! jix! jiy! jiz inertia matrix J ji diag ( J jix (! ) = J jiy # ] T. The angular velocity! and moment of are oth determined in ody frame, J ji = J jiz ). J jiz! jiz -J jiz! jiz J jiy! jiy -J jiy! jiy J jix! jix -J jix! jix The spatial revolute allows turning the odies joined etween them. Therefore, three translations and two rotational degrees of freedom are constrained, leaving only one rotation degree of freedom free. For a single moving chain j (j=,,, representing chain A-a, B-, or C-c respectively), the vector ond graph model can e made y assemling the % & ji vector ond graph of a single spatial moving rigid ody [], the revolute s, and the universal, which is shown in Fig. (6). In Fig. (6), if i=a and j=, it presents the vector ond graph of moving chain A-a, if i= and j=, it presents the vector ond graph of moving chain B-, and if i=c and j=, it presents the vector ond graph of moving chain C-c. &r C ji (i=,,, j=,,) is the mass center velocity vector of ody i of chain j in gloal coordinates,! ji the angular velocity vector of ody i of chain j in ody frame,! ji is the angular velocity vector of ody i of chain j in gloal coordinates. The mass of ody i is m M i =diag ( m m m Ci ). The vector C i C i C i ond graph shown in Fig. (6) can e simplified and shown in Fig. (7), where pi and ip are two output ports which connect the vector ond graph models of three moving chains (A-a, B-, and C-c) to upper platform vector ond graph model ( i=a,, c ). The upper platform is a spatial moving rigid ody, its vector ond graph model is shown in Fig. (8), where r & up is the mass center velocity vector of upper plat- form in gloal coordinates, The angular velocity vector! up

5 A Computer Aided Kineto-Static Analysis Method The Open Automation and Control Systems Journal,, Volume 5 7 Chain A-a ap pa ap pa Upper platform p p p Chain B- p pc cp Fig. (7). The simplified model of Fig. (6). and moment of inertia matrix J p ody frame, J p = diag ( platform is ap pa MTF Ø up a8 p 8 c8 Se I: J p p MTF r up J px I:M p J py J MTF Fig. (8). The vector ond graph model of upper platform. ap pa are oth determined in pz m pc, M p =diag ( m pc m pc m pc ). ), the mass of upper The vector ond graph model of upper platform can e simplified and shown in Fig. (9), where ap, p, cp, pa, p, pc present the output ports of the vector ond graph mode of moving chains ( A-a, B-, C-c ) respectively. p p Upper platform pc Fig. (9). The simplified vector ond graph model of upper platform. cp pc p p cp As a result, the overall vector ond graph model of - RRRT manipulator can e made and shown in Fig. (). However, the kinematic constraints result in differential causalities, the current vector ond graph procedures [9-] were found to e very difficult algeraically in deriving system driving moment (or force) and the constraint force equations automatically on a computer. To solve this prolem, the constraint force (or moment ) vectors of s can e considered as unknown source vectors, such as Se i i, Se i 4 i 7 i 9 i i4 i7 ( i=a,,c ), and added to the corresponding -junctions. As a result, all differential causalities in this system vector ond graph can e eliminated, thus the procedure presented here can e used directly. 4. THE UNIFIED FORMULAE OF DRIVING MO- MENT AND CONSTRAINT FORCES FOR SPATIAL ROBOT SYSTEMS The asic fields and junction structure of system ond graph is shown in Fig. () [], where Euler-junction structure (EJS) [, ] is added. X i represents energy variale vector of independent storage energy field corresponding to independent motion, X i represents energy variale vector of independent storage energy field corresponding to dependent motion, Z i and Z i are the corresponding coenergy variale vectors. D in and D out represent input and output vector variales in resistive field, U and V represent input and output vector variales of source field respectively, U =! U U U! V V V T #, V = T #. Where U is driving moment (or force) vector, U is the constraint force vector of, and U is known source vector. E in and E out are the input and output variale vectors in Euler-junction structure (EJS). For independent energy storage field, we have pc Z i = F i X i (4) Z i = F i X i (5) where F i and F i are m! m and m! m matrices respectively. For resistive field, we have Chain C-c Fig. (). The overall vector ond graph model of -RRRT manipulator. cp

6 8 The Open Automation and Control Systems Journal,, Volume 5 Dout = RDin (6) Wang and Tao A4 = J Lu + J ELR (I! J LLR )! J Lu where R is L! L matrix. The EJS equations of a overall multiody system can e assemled y Eq.() and written as Eout = RE Ein (7) n L Ein ]T, RE = diag(re n A6 = J Eu + J ELR (I! J LLR )! J Lu B = (I! J LLR )! (J Li Fi + J LE AA ) B = (I! J LLR )! (J Li Fi + J LE AA ) RE L RE ). A5 = J Eu + J ELR (I! J LLR )! J Lu Eout L Eout ]T, E in = [Ein Ein where E out = [Eout n The corresponding junction structure equations can e written as B = (I! J LLR )! (J Lu + J LE AA4 ) X& i = J i i Z i + J i i Z i + J i L Dout + J i u U B 4 = (I! J LLR )! (J Lu + J LE RE AA5 ) +J i u U + J i u U + J i E E out (8) (9) () () () By the algeraic manipulation from Eq.(4)~Eq.(), the system driving moment and constraint force equations can e written as H = J&Ci Fi + J Ci Fi Ti i + J Ci Fi Ti i +Ti Tu X& i + Ti Tu Ti u H 4!J cu U& ) H = J&Ci Fi + J Ci Fi Ti i + J Ci Fi Ti i H = J Ci Fi Ti u + J Ci Fi Ti u H 5 = J&Cu + J Ci Fi Ti u + J Ci Fi Ti u! S u u = T [Ti u + Ti u (!H 4 ) H ] T iu U = S u!u (S u i X i + S u i X i + S u u U S u i = Ti Tu (Ti u H 4!H! Ti i ) J CL =, J CE = H 4 = J Ci Fi Ti u + J Ci Fi Ti u +J CE E out If = J Ci Z i + J Ci Z i + J CLD out + J Cu U Ti u = J i LRB 4 + J i u + J i E AA5 From the flow summation of -junctions corresponding to m constraint force vectors in system vector ond graph model, we have Ti u = J i LRB5 + J i u + J i E AA6 +J Eu U + J Eu U + J EE E out Ti i = J i i Fi + J i L RB + J i E RE A A E in = J Ei Z i + J Ei Z i + J ELD out + J Eu U Ti u = J i L RB + J i u + J i E RE A A4 +J Lu U + J Lu U + J LE E out D in = J Li Z i + J Li Z i + J LLD out + J Lu U B5 = (I! J LLR ) (J Lu + J LE AA6 ) +J i u U + J i u U + J i E E out! Ti i = J i i Fi + J i L RB + J i E RE A A X& i = J i i Z i + J i i Z i + J i LD out + J i u U (a)! U = (!H 4 ) (HX i + H X i + H U +H 5U + J cu U& ) () # % S u i = Ti Tu (Ti u H 4!H! Ti i ) S u u = Ti Tu (Ti u H 4!H 5! Ti u ) () If J CL! or J CE! U = D u!u (D u i X i + D u i X i + D u u U +Ti Tu X& i ) where (a) A = [I! J ELR (I! J LLR )! J LE! J EE ]! U = (!TCu )! (TCi X i + TCi X i + TCu U A = J Ei Fi + J ELR (I! J LLR )! J Li Fi +TCu U ) A = J Ei Fi + J ELR (I! J LLR )! J Li Fi where () # % (4)

7 A Computer Aided Kineto-Static Analysis Method The Open Automation and Control Systems Journal,, Volume 5 9 Fig. (). The asic field and junction structure of system. T Ci = J Ci F i + J CL RB + J CE A A T Ci = J Ci F i + J CL RB + J CE A A T Cu = J CL RB + J CE A A 4 T Cu = J CL RB 4 + J CE A A 5 T Cu = J CL RB 5 + J Cu + J CE A A 6 D u u = T i u D u i = T i u D u i = T i u D u u = T i u T [T i u +T i u (-T Cu ) - T Cu ] T (T i T - u Cu T Ci - T i ) i T (T i T - u Cu T Ci - T i ) i T (T i T - u Cu T Cu - T i ) u Giving the system independent moving state variale vector X i and its derivative X & i, the corresponding system driving moment (or force) vector U and constraint force vector U can e determined from Eq.() or Eq.(4) directly. 5. KINETO-STATIC ANALYSIS EXAMPLE To check the model and procedure presented here, the kineto-static analysis for -RRRT manipulator is carried out. For chain j (j=,,), the mass of ody i (i=,,) is M ji, M j =kg, M j =.5kg, M j =kg. The length of ody i (i=,,) in chain j (j=,,) is presented L ji, L j =.4m, L j =.m, L j =.8m. R=.4, r=.m. The moment of inertia matrix is J ji. J ji = #! L! L! L The mass center motion of upper platform are as following, x =.5cos(.5! t ),.5sin(.5! t) z =.8. up % & y up =, up From Fig. (), vector X i, X i, Z i, Z i, D in, D out, U, U, U, V, V, V, D in, D out, E in and E out can e defined. By Eq.(4) ~Eq.(7), the matrix F i, F i, R and can e otained. Also the coefficient matrices of Eq. (8)~Eq.() which are called junction structure matrices can e got from Fig. (). In this example, ecause the motion of upper platform is in a plane, so that the corresponding EJS can e eliminated shown as Fig. (8). Inputting the physical parameters of the manipulator, the matrix F i, F i, R and junction structure matrices in Eq.(8)~Eq.(), known source vector U, system independent moving state variale vector X i, and its derivative & X i into the program associated with the procedure presented here ased on MATLAB [], the system driving moment (or force) and constraint force equations in the form of Eq. (4) can e derived on a computer, and the corresponding driving moment (or force) and constraint forces can e determined. Some of results are shown in (Fig. to 7). For this example, the Newton-Euler method [] was used to determine the corresponding driving moment (or force) and constraint forces, the results are in good agreement with that otained y the procedure in this paper. However, this process is very laor-intensive and tedious.

8 The Open Automation and Control Systems Journal,, Volume 5 Wang and Tao Fig. (). The driving moment of chain A-a. Fig. (). The driving moment of chain B-. Fig. (4). The driving moment of chain C-c.

9 A Computer Aided Kineto-Static Analysis Method The Open Automation and Control Systems Journal,, Volume 5 Fig. (5). Resultant constraint force etween upper platform and ody in chain A-a. Fig. (6). Resultant constraint force etween ase platform and ody in chain A-a. Fig. (7). Resultant constraint force etween ody and ody in chain A-a. From the descriptions aove, the method presented here is suitale for oth open loop root mechanism and closed loop root mechanism. Besides this, the method is not required mechanism acceleration analysis at all. Thus the reliaility and efficiency of the kineto-static analysis of complex root systems can e improved y the procedure presented here.

10 The Open Automation and Control Systems Journal,, Volume 5 Wang and Tao 6. CONCLUSIONS The vector ond graph procedure presented here is very suitale for dealing with computer aided kineto-static analysis of complex root systems with the coupling of multienergy domains. Compared with traditional scalar ond graph method, this vector ond graph procedure is more suitale for complex spatial root mechanism ecause of its more compact and concise representation manner. The differential causalities in the vector ond graph model of spatial root mechanisms can e avoided y the ond graph augment method proposed here, thus the algeraic difficulties in system automatic modeling and kineto-static analysis can e overcome. In the case of considering EJS, the unified formulae of system driving moment and constraint force equations are derived, which are easily derived on a computer in a complete form. Besides these, the procedure is suitale for oth open loop root mechanism and closed loop root mechanism and not required mechanism acceleration analysis at all. These lead to a more efficient and practical automated procedure for kineto-static analysis of complex root systems over a multi-energy domains in a unified manner. The validity of the procedure is illustrated y successful application to the kineto-static analysis of -RRRT manipulator. CONFLICT OF INTEREST The authors confirm that this article content has no conflicts of interest. ACKNOWLEDGMENTS This work was financially supported y National Natural Science Foundation of China (Grant No. 5757). REFERENCES [] J. Z. Hong, Computational Dynamics of Multiody Systems, The Advanced Education Press, Beijing,. [] Z. Huang, L. F. Kong and Y. F. Fang, The parallel root mechanism theory and control, Machine Press, Beijing, 997. [] D. C. Karnopp, D. L. Margolis and R. C. Rosenerg, System dynamics: modeling and simulation of mechatronic systems, 4th ed., John Wiley, New York, 6 [4] R. C. Rosenerg, Reflections on Engineering Systems and Bond Graphs, ASME, J. Dyn. Syst, Meas. Control, vol. 5, no., pp. 4-5, 99. [5] D. C. Karnopp, Understanding multiody dynamics using ond graph representations, J. Franklin. Inst., vol. 4B, no. 4, pp. 6-64, 997. [6] D. L. Margolis and D. C. Karnopp, Bond Graph for Flexile Multiody Systems, Trans. ASME Ser. G, vol., no., pp. 5-5, 979. [7] A. A. Zeid, J. L. Overholt, Singularly pertured ond graph models for simulation of multiody systems, J. Dyn. Syst., Meas. Control., vol.7, no., pp. 4-4, 995. [8] Zeid, C. H. Chung, Bond graph modelling of multiody systems: a lirary of three-dimensional s, J. Franklin. Inst., vol. 9, no. 4, pp , 99. [9] J. Jinhee, H. Changsoo, Proposion of a modelling method for constrained mechanical systems ased on vector ond graphs, J. Franklin. Inst., vol. 5B, no., pp , 998. [] F. Wilfird, S. Serge, Bond Graph Representation of Multiody Systems with Kinematic Loops, J. Franklin. Inst., vol. 5, no. 4, pp , 998. [] S. Behzadipour, A. Khajepour, Causality in vector ond graphs and its application to modeling of multi-ody dynamic systems, Simul. Modell. Prac. Theory, vol. 4, pp , 6. [] P. Breedveld, Staility of Rigid Rotation from a Bond Graph Pespective, Simul. Modell. Prac. Theory, vol.7, pp. 9-6, 9. [] J. F. Jiang, L. J. Hu and J. Tang, Numerical Analysis and MATLAB Experiment, Science Press, Beijing, 4 Received: August, Revised: Septemer 8, Accepted: Septemer 8, Wang and Tao; Licensee Bentham Open. This is an open access article licensed under the terms of the Creative Commons Attriution Non-Commercial License ( which permits unrestricted, non-commercial use, distriution and reproduction in any medium, provided the work is properly cited.