methods for tearing systems of equations in object
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1. B a o NEWx3 This graph corresponds to the following successive substitution algorithm NEWx2 INITX2 NEWx3 INITX3 REPEAT x2 NEWx2 x3 NEWx3 xl f1INV1 x2 x3 NEWx2 f2INV2 x1 x3 NEWx3 f3INV3 x1 x2 UNTIL converged NEWx2 x2 AND converged NEWx3 x3 iINVj denotes the function obtained when solving f 0 for Xi This algorithm is similar to the one obtained when tearing the directed graph There is however a second alternative to tear a branch in a bipartite graph as illustrated below ON lan f lt oe a oP mee OLDx CALCx The original graph could for example be torn by this method in the following way fl Rearranging this loop free graph gives a calculation order CAROK O Seto tras yaaa i gt OLDx2 CALCx3 gt OLDx3 fl C xl f2 x2 f3 x3 This corresponds to the following algorithm successive substitution x2 INITX2 x3 INITX3 REPEAT OLDx2 x2 OLDx3 x3 xl f1INV1 OLDx2 OLDx3 x2 f2INV2 x1 OLDx3 x3 3INV3 x1 x2 UNTIL converged x2 OLDx2 AND converged x3 OLDx3 Note that x3 is calculated using the new value of x2 instead of OLDx2 This is the difference compared to the previous algorithm For iterative solution of linear systems of equations the last method corresponds to Gauss Seidel iteration and the previous method to Jacobi iteration2. The appropriate sign of must be chosen in order that the differential equation would have a stable solution Solving algebraic loops in this way leads to stiff equations A notation to make this infinitely stiff is needed i e a notation to state that a solution should not be obtained by an integration method but by a solver for simultaneous equations The following notation is used residue x f x residue is a special operator just like der the Dymola language construct for derivative The value of residue x is always maintained as zero The model however gives a hint about what to vary x to make residue x zero The equation will be associated with the generated identifier residuex which may be used in the variable tear command If an equation containing residue x is differentiated der x is considered a candidate as tearing variable i e der residue x residue der x Normal use of residue x introduces both a tearing variable and a residue equation This allows to create pairs which are for example needed in order to keep a possible symmetry property of the matrix H above The notation allows to specify just a tearing variable by adding residue x to any equation which can never be part of a system of simultaneous equations Correspondingly an equation is marked as a residue equation by adding residue c to it where c is declared as a unique constant To summarize Dymola will take th
3. E mail df43 master df op dlr de linear and non linear systems can be utilized to solve the reduced systems of equations PRINCIPLES OF TEARING There are different concepts of tearing depending on whether tearing is done before or after the computational causality has been determined i e transformation from general equations to assignment statements Cellier and Elmqvist 1993 Tearing in directed graphs Consider the following system of simultaneous equations x fia X2 ha The structure of these equations can be represented by the directed graph shown below fl xl x2 P The mutual dependency in the equations gives a loop in the graph an algebraic loop The loop can be removed by tearing the graph apart for example breaking the branch corresponding to x2 This gives the following dependency graph gt fl gt f2 x2 xl NEWx2 This graph suggests the following way of organizing the calculations in a successive substitution algorithm for finding the solution of the equations NEWx2 INITx2 REPEAT x2 NEWx2 x1 1 x2 NEWx2 2 x1 UNTIL converged NEWx2 x2 It should be noted that the iteration is only performed for x i e the dimension of the problem has been reduced from two to one The fixed point iteration scheme can easily be replaced by Newton iteration such that x is the only unknown variable and x2 f2 fi x2 0 is the non linear
4. Node tearing in bipartite graphs Instead of cutting single branches in the graph it is possible to cut all branches to a particular node and to remove the node Duff 1986 Alternatively it can be seen as making additions Removing a variable node has the same effect as adding an equation to specify the variable Removing an equation node is the same as relaxing the equation by including a residue variable Consider again the equations Ait x2 x3 0 f x x2 3 0 A x x2 x3 0 By specifying x2 and x3 as tearing variables and adding residues to f2 and f3 we get the following rearranged graph CALCx2 5 This graph corresponds to the algorithm below for calculating the residues x2 XS SAA xI 1INV1 x2 x3 2 xiL x2 X3 ae O a E 2 E An advantage with this method is that no inverses are required for f2 and f3 It is well suited for Newton iteration and for solving linear systems of equations as shall be seen below This scheme does not utilize causality assignment information On the other hand the user must specify what equations to use as residue equations Only node tearing will be considered below SOLVING TORN NON LINEAR SYSTEMS OF EQUATIONS The node tearing problem can be formulated as follows Elmqvist 1978 Find a partitioning of the equations f and the variables x and permutation matrices P and Q such that Pf h y x Q Z The system of equations f x 0 can
5. facilities makes it possible to treat these kind of models in a uniform way REFERENCES Cellier F E and H Elmqvist 1993 Automated Formula Manipulation Supports Object Oriented Continuous System Modeling IEEE Control Systems Vol 13 No 2 Duff I S A M Erisman and J K Reid 1986 Direct Methods for Sparse Matrices Oxford Science Publications Elmqvist H 1978 A Structured Model Language for Large Continuous Systems Ph D thesis Report CODEN LUTFD2 TFRT 1015 Department of Automatic Control Lund Institute of Technology Lund Sweden Elmqvist H 1994 Dymola User s Manual Dynasim AB Research Park Ideon Lund Sweden Kron G 1963 Diakoptics The Piecewise Solution of Large scale Systems MacDonald amp Co London Luh J Y S M W Walker and R P C Paul 1980 On line computational scheme for mechanical manipulators Trans ASME Journal of Dynamic Systems Measurement and Control Vol 102 pp 69 76 Mah R S M 1990 Chemical Process Structures and Information Flows Butterworths Otter M H Elmqvist and F E Cellier 1993 Modeling of Multibody Systems With the Object Oriented Modeling Language Dymola In Proceedings of the NATO ASI Computer Aided Analysis of Rigid and Flexible Mechanical Systems Troia Portugal Vol 2 pp 91 110 Schiehlen W 1993 Advanced Multibody System Dynamics Simulation and Software Tools Kluwer Academic Publishers Tarjan R E 1972 Depth F
6. at every mechanical object and auxiliary variables For example an object oriented model of a typical robot with 6 revolute joints leads to a sparse linear system of equations with about 600 equations Below it is shown that appropriate tearing transforms these huge systems of equations into small systems of equations which correspond to the usual forms derived by mechanical principles like Lagrange s equations or Kane s equations The tearing information can be given in the class library for mechanical systems such that no user interaction is required i e the user need not be aware of the underlying tearing procedure Tearing for mechanical systems will first be explained for tree structured mechanical systems and afterwards for general systems containing kinematic loops Tree structured mechanical systems Mechanical systems are called tree structured when the connection structure of bodies and joints forms a tree Typical examples for tree structured systems are robots or satellites Many formalisms are known see e g Schiehlen 1993 to derive the following standard form of tree structured systems M q q h q q f 1 where q are the relative coordinates of all joints and f are the known generalized forces in the joints e g the torque along the axis of rotation of a revolute joint produced by an electric motor This equation can be easily transformed to state space form by solving 1 for q and by usin
7. equation to be solved for In this example the causalities of the equations are given This is the typical problem formulation within for example static simulation in chemical engineering see e g Mah 1990 Branch tearing in bipartite graphs Equations without assigned causality will now be studied Consider the following equations fiX X 3 0 A X X xX 0 BO Xp xX 0 This system of equations can be represented by a bipartite graph Tarjan 1972 i e a graph that has two sets of vertices one set of vertices for equations and one for variables fl Tearing a branch in a bipartite graph between an equation f and a variable x can be done in two ways since tearing also implies assigning causality to the torn branch The first case means that x gets defined by a new equation CALCx and that x is replaced by NEWx in the equation f The corresponding new graph is or in a compact notation Assume that the branches f2 x2 and f3 x3 in the graph above are torn in this way fl The resulting graph does not contain any loops This is the usual criterion that tearing is successful It is thus possible to rearrange the graph corresponding to ordering calculations to be performed sequentially i e downwards in the graph CALCx2 x2 CALCx3 C x3 fl O xl f2 NEWx2
8. it is possible to calculate the node voltages without solving any systems of equations Introducing known mesh currents can be seen as introducing small inductors in those branches since the current of an inductor is a state variable i e known A model class with an infinitely small inductor is described as follows model class TwoPin MeshCut Corresponds to very small inductor u residue i Cf normal inductor u der i L end Three MeshCuts are needed to tear the circuit i e to reduce the number of algebraic equations from 15 to 3 They are connected as follows submodel MeshCut MC1 MC2 MC3 connect Common UO R1 MC1 R2 R3 MC2 R4 R5 MC3 R6 Common The corresponding number of operations needed in this case are 28 MULT and 25 ADD i e a speed up by a factor 4 5 An alternative is to introduce node voltages and formulate equations for calculating the currents Introducing node voltages correspond to connecting small capacitors from the nodes to ground model class NodeCut Corresponds to very small capacitor connected to ground main cut A V i i residue V i der v c end Adding nodecuts for u uy and ug gives the corresponding operation counts 38 MULT and 25 ADD These methods have traditionally been used for circuit analysis Inspecting this particular circuit gives other possibilities however Assuming that the current i was known it is po
9. just used in order to define the angle q and the angular derivative qd as tearing variables Since tearg tearqd do not appear in any algebraic loop the corresponding residue equations e g tearg residue q are just ignored The last equation is already known for the tearing of tree structured systems As an example for cut joints the spherical joint j2 of the four bar mechanism is discussed This joint type contains the following equations in its class description constant dummyq 3 0 dummyqd 3 0 local fc 3 rrel 3 vrel 3 arel 3 residue dummyq rrel residue dummyqd vrel residue fc arel When a cut joint is removed two cut planes are present called cut a and cut b respectively The relative vector from cut a to cut b is denoted as rrel It is calculated from the kinematic information of the two objects attached at cut a and cut b respectively For a spherical joint this relative vector must be zero This equation is therefore used as residue equation of the joint at position level Since dummyq is a constant and therefore known it is not used as a tearing variable In the same way the second equations states that the relative velocity vrel must be zero and is used as residue equation Finally the last equation states that the relative acceleration arel must be zero and is also used as residue equation Furthermore the constraint forces fc of the spherical joint are used as tearing variabl
10. then be written as g y z 0 h y z 0 The criterion for the partitioning and permutation is usually to make the Jacobian as lower triangular and the dimension of z as small as possible The i th equation of g is then independent Of Yip Yny i e it can be written as BIO lt Yis Us Znz 0 In many cases the equation is linear in y it can then be solved symbolically for y by rearranging as BOI Vids Zb Znz uO lt Yi Ze Znz yi O This gives a method for successively solving for all components of y y 8 z Substituting y in the h equations gives h g z z 0 or H z 0 i e a non linear system of equations in only z is obtained This can for example be solved by Newton Raphson iteration Given the tearing variables z and the residue equations h the permutation matrices P and Q can be determined by efficient algorithms In particular the same algorithms can be used as utilized for the equation and variable sorting of the original object oriented model to determine e g algebraic loops of minimal dimensions Dymola forms g symbolically as explained above The residues h are trivially obtained In addition Dymola forms the Jacobian mM MZ M OL OY oa a by differentiating symbolically and makes call to numerical routines for solving H z 0 iteratively SOLVING TORN LINEAR SYSTEMS OF EQUATIONS If the system of equations is linear in x it is transformed to bordered triangular form i e t
11. METHODS FOR TEARING SYSTEMS OF EQUATIONS IN OBJECT ORIENTED MODELING Hilding Elmqvist Dynasim AB Research Park Ideon S 223 70 Lund Sweden E mail Elmqvist Dynasim se ABSTRACT Modeling of continuous systems gives a set of differential and algebraic equations In order to utilize explicit integration routines the highest order derivatives must be solved for In certain cases there exist algebraic loops i e subsets of the equations must be solved simultaneously The dependency structures of such subsets are often sparse In such cases the solution may be found more efficiently by a technique called tearing Kron 1963 which reduces the dimensions of sub systems This paper gives an overview of the principles of tearing Algorithms to determine how a set of equations should be torn are in general inefficient However physical insight often suggests how this should be done Methods to specify tearing in the object oriented modeling program Dymola Elmqvist 1978 1994 are discussed In particular it is explained how tearing can be defined in model libraries This allows Dymola to perform tearing automatically and efficiently without user interaction Examples from electrical and mechanical modeling are given including a tearing strategy for general multibody systems with kinematic loops which allow the equations of motion to be solved by standard explicit integration algorithms INTRODUCTION In object oriented modeling comput
12. e following steps when testing if tearing should be performed when a system of simul taneous equations is processed e Check for automatic tearing Among unknowns select as tearing variables those that residue has been used on Select as residue equations those that contain residue e Add or subtract user defined tearing variables and residue equations Since the user might have to add more variables or vice versa the command must allow adding a variable only or an equation only The syntax of the variable tear command is thus extended with the two cases variable tear v Add v as a tearing variable but no equation variable tear residuev Add residue equation with residuev EXAMPLES Electrical circuits Systems of simultaneous equations occur for example in a voltage divider consisting of two resistors Consider the generalization below R1 R3 R5 i1 u2 i3 u4 i5 u6 gt gt 7 U E R2 R4 R6 i2 V i4 V i6 A system of 15 equations is detected trivial equations not counted If the symbolic solver is used the operation counts are totally 126 MULT and 93 ADD One alternative way of formulating the governing equations of such a network is to find a spanning tree and for branches not included in that tree introduce mesh currents see e g Vlach and Singhal 1994 Assuming these mesh currents are known
13. er models are mapped as closely as possible to the corresponding physical systems Models are described in a declarative way i e only local equations of objects and the connection of objects are defined The problem formulation determines which variables are known and unknown As a result an object oriented model usually leads to a huge set of differential and algebraic equations Efficient graph theoretical algorithms are available to transform these equations to an algorithm for solving the unknown variables usually the highest order derivatives see e g Duff et al 1986 Mah 1990 Elmqvist 1978 Especially algebraic loops of minimal dimensions can be determined Tarjan 1972 For models of physical systems like electrical circuits or mechanical systems algebraic loops of minimal dimensions are often quite large Since the model equations are derived from an object oriented model description the systems of equations are usually very sparse i e only few variables are present in an equation In such cases the solution may be found more efficiently by reducing the dimensions of the systems of equations by a technique called tearing Kron 1963 This method has the advantage that the system reduction can be done symbolically and that general purpose numerical solvers for Martin Otter Institute for Robotics and System Dynamics German Aerospace Research Establishment Oberpfaffenhofen DLR Postfach 1116 D 82230 Wefling Germany
14. es Note that for mechanisms with kinematic loops the tearing variables and the corresponding residue equations are not defined in the same joint class Instead the tearing variables at position and velocity level are defined in the joint classes for the spanning tree whereas the corresponding residue equations are defined in the cut joint classes CONCLUSIONS Connecting models introduces constraints Such constraints may lead to the requirement to solve systems of simultaneous equations while computing the derivatives Users normally introduce auxiliary variables for common subexpressions Such variables however increase the dimension of the system of equations and makes it less efficient to solve It is however sparse This fact is utilized in the tearing method Tearing in fact removes auxiliary variables from the set of variables being iterated Different principles of tearing and how it is utilized have been described Notations to specify tearing in the object oriented language Dymola have been given Important applications are modeling of electrical circuits and 3D mechanical systems In particular it was shown that for quite general mechanical systems tearing can be defined in the mechanical class library i e the user must not be aware of the underlying tearing procedure Traditionally special purpose simulators are used for such systems The object oriented language Dymola with its tearing and symbolic manipulation
15. for Qmins Arest The discussed equations can be generated from an object oriented model by using an appropriate type of tearing For this the user has to split up the joints of a mechanism into 3 categories cut joints state variable tree joints and remaining tree joints As already explained the removal of all cut joints will result in a tree structured system A state variable tree joint is a joint of the spanning tree where the relative coordinates of the joint and their first derivatives are used as state variables The remaining tree joints are the remaining joints of the spanning tree For example in the four bar mechanism spherical joint j2 is used as cut joint revolute joint jZ is used as state variable tree joint and the cardan and prismatic joints j3 j4 are used as the remaining tree joints Note that this mechanism has one degree of freedom and that the angle and the angular velocity of joint j1 are used as state variables due to the selection of j as state variable tree joint The variables Qmin Gmin i the coordinates of the state variable tree joints are known because these quantities are used as state variables Assuming that Qyest Qrest i the coordinates of the remaining tree joints as well as the constraint forces f of the cut joints would be known the equations of motion could be determined since this system has all the properties of a tree structured s
16. g q and q as state variables By Luh et al 1980 it was shown that the generalized forces f of tree structured mechanical systems can be calculated given q q q without encountering algebraic loops solution of the inverse dynamics problem in robotics This means that the generalized forces f and all other interesting quantities can be determined in a sequential manner given the known state variables q q and the generalized accelerations q As a consequence equation 1 can be generated from an object oriented model by using the unknown generalized accelerations q as tearing variables and the equations to compute the generalized forces f as residue equations It can be shown that this type of tearing leads to an O n algorithm to compute 1 where n is the dimension of q In particular this algorithm is equivalent to algorithm 1 of Walker and Orin 1982 Note that another O n operations are needed to solve 1 for q Therefore the overall algorithm is O n3 For the mentioned robot with six revolute joints the explained type of tearing reduces the number of equations from 600 to 6 i e to equation 1 The complete tearing information can be given in the joint classes E g for a revolute joint the following equation is present in the class description f nl tl n2 t2 n3 t3 residue qdd Here f is the generalized force of the revolute joint i e the known torque along the axis of rotation n n1
17. gebraic equations M q q h q q f G Qf 2a og 0 g g q G 2b s 2 qg a 0 G q q 2c 0 8 G qq EV 2d oq Here 2a are the equations of motion of the spanning tree mechanism 2b 2c 2d are the constraint equations of all cut joints at position velocity and acceleration level respectively q are the relative coordinates of the joints of the spanning tree f are the known generalized applied forces of the joints of the spanning tree and f are the unknown generalized constraint forces of the cut joints In a second step state variables must be selected i e for every kinematic loop 6 n position and 6 n velocity coordinates of the joints of the spanning tree must be defined as unknown where n is the number of degrees of freedom of the cut joint of the loop The joint coordinates of the spanning tree are therefore split up into q Qmin Qrest Where qmin are the known position state variables and Qyes are the unknown remaining position coordinates Qmin are the velocity state variables The derivatives of the remaining coordinates qyest rest are treated as algebraic variables by the numerical integration algorithm Given the state variables qmin s Amin the derivatives of the state variables qmin can be calculated from 2 in the following way 2b is a non linear system of equations for qyes 2c is a linear system of equations for qrest and 2a 2d is a linear system of equations
18. however important with good diagnostics to handle the case when the specification is inconsistent does not give a complete tearing or alternatives due to solvability need to be tested Two sets are needed for node tearing the tear variables and the residue equations They don t have to be specified as pairs but by proper pairing the matrix H for linear systems of equations might become symmetric giving a more efficient solution The specification of tearing variables poses no problem because in an object oriented model every variable is uniquely identified by its hierarchically structured name The remaining problem is then how to specify residue equations since they don t have names or numbers When a system of equations appears the user might have to look at the structure of the equations in order to propose tearing variables and residue equations For small systems he might include a listing of the incidence matrix At that time there is an assignment between variables and equations In the EQUATIONS section of the output from Dymola the assigned variable is marked by This association actually means that every equation can be referred to by name i e the assigned variable name The tearing specification could thus be done by a translator command like variable tear tear_variable residue_equation_ variable Both specified variables should be unknown in the system of equations under consideration A natural default if
19. irst Search and Linear Graph Algorithms SIAM J of Comp 1 pp 146 160 Walker M and D Orin 1982 Efficient Dynamic Computer Simulation of Robotic Mechanisms Trans ASME Journal of Dynamic Systems Measurement and Control Vol 104 pp 205 211 Vlach J K Singhal 1994 Computer Methods for Circuit Analysis and Design Second Ed Van Nostrand Reinhold
20. n2 n3 is a unit vector in direction of the axis of rotation t t1 t2 t3 is the cut torque and qdd is the second derivative of the angle of rotation q The equation states d Alembert s principle i e that the projection of the unknown constraint torque t onto the axis of rotation n is the known applied torque As already explained the term residue qdd states that the equation is used as residue equation and that qdd is used as tearing variable Mechanical systems with kinematic loops A simple mechanism with one kinematic loop is shown in the figure below One difficulty with such types of systems is that the relative joint coordinates are no longer independent from each other because the kinematic loops introduce additional constraints As a consequence only a subset of the relative joint coordinates can be selected as state variables spherical joint G2 Cardan joint revolute joint G1 prismatic joint G4 Mechanical systems with kinematic loops can be handled by cutting selected joints such that the resulting system has a tree structure The removed cut joints are thereby replaced by appropriate unknown constraint forces and torques Further more the kinematic constraint equations of the cut joints are added as additional equations to the equations of motion of the tree structured system It is well known see e g Schiehlen 1993 that this procedure leads to the following system of differential al
21. o the following form L Ap Y E b Ay Ao Z b where L is lower triangular and the dimension of z low The inverse of L can be found efficiently without pivoting y can thus be expressed as 1 y L b A pZ Substituting into the second equation gives Az2 An L An z b2 AzL b Introducing H Az2 AzL A12 c b2 Ay L by gives Hz c In Dymola the matrix H and the vector c are formed symbolically The solution of z from Hz c is either obtained symbolically or by calls to numerical routines The remaining variables y are then solved symbolically from the original equations SELECTION OF TEARING VARIABLES AND RESIDUE EQUATIONS Automatic selection of tearing variables and residue equations is algorithmically hard For an overview of heuristic algorithms see Mah 1990 A good selection must also take into account how difficult it is to solve certain equations linear non linear etc Solution of certain linear non residue equations g might require division by a parameter Such a selection might not be feasible because it is then not possible to set the parameter value equal to zero at simulation time Convergence properties both for successive substitution and Newton iteration are also influenced by the tearing selection For use of Newton iteration the complexity of the Jacobian is also influenced Physical insight on the other hand might suggest how to make tearing If a tearing specification is made manually it is
22. ssible to calculate u U R i giving i uy R iz i i Similarly it would be possible to calculate i and ig These should be identical thus i ig is the residue for i This scheme can be formulated using the following model class model class TearCut cut A V i B V i main path P lt A B gt cut Res Vres ires ires residue i end The main path is connected in series with R1 The residue cut is connected at the node having voltage u6 The resulting operation counts are 27 MULT and 25 ADD i e the best of the three alternatives considered Mechanical systems Three dimensional mechanical systems can be described in an object oriented way with Dymola see Otter et al 1993 for details Here physical objects like bodies joints or force elements are defined as objects of corresponding Dymola classes These objects are connected together according to the physical coupling of the components of a mechanism As generalized coordinates q the relative coordinates of joints are used e g the angle of rotation of a revolute joint It turns out that such an object oriented description of mechanical systems leads to huge systems of algebraic equations containing the following unknown quantities the absolute acceleration and the absolute angular acceleration of every frame coordinate system the second derivatives of the relative coordinates of every joint the cut forces and cut torques which are present
23. the second variable is omitted is that the residue_equation_variable is the same as the tear_variable TEARING INFORMATION IN THE MODEL Several ways of specifying tearing are needed The one introduced has the advantage of being general and no prepara tions changes to libraries are needed The drawback is that the user has to output the solved equations in order to analyze the systems of equations before choosing tearing variables and residue equations For standard problems this step should be avoided In such a case it is better that preparations are made to the library It is a matter of giving Dymola hints about possible tearing variables and residue equations It should be hints in the sense that if no systems of equations appear the information is just ignored It would be nice if cuts of the dependencies could be defined among the objects instead of among the variables Examples are defining cuts in closed loop mechanical mechanisms or electrical circuits Introduction of cut objects would give a more high level and intuitive notation One way to reason in order to design appropriate notations is to consider why algebraic loops occur They are often caused by neglected dynamics i e dynamics considered so fast that steady state occurs immediately Consider the scalar equation O f x A solution to this equation could be obtained by solving the differential equation dx _ E IN using a small value of
24. ystem Therefore yest Arest f must be used as tearing variables in addition to q which are already known to be tearing variables in order to tear the spanning tree mechanism The residue equations are just the constraint equations of the cut joints at position velocity and acceleration level in addition to the already known residue equations for the calculation of the generalized forces f of the joints of the spanning tree For the four bar mechanism the object oriented model results in 3 distinct systems of equations A system of 123 non linear equations at position level a system of 93 linear equations at velocity level and a system of 182 linear equations dynamic equations The discussed tearing procedure reduces the dimensions of these systems of equations considerably to a system of 3 non linear equations at position level a system of 3 linear equations at velocity level and a system of 7 linear equations 4 dynamic equations of the spanning tree and 3 constraint equations on acceleration level In a similar way as for pure tree structured systems the complete tearing information can be given in the joint classes E g for a revolute joint which is used as a joint in a spanning tree the following equations are present in the class description local tearg tearqd tearq residue q tearqd residue qd f nl tl n2 t2 n3 t3 residue qdd The two local variables teargq tearqd are dummy variables and
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