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Mc cBILE Version 1.3 User's Guide

1. 18 The two types of algebraic scalar objects in M JBILE 22 Class hierarchy for the different types of matrices in MTJBILE 25 State subentries of state objects o o oo a ee 28 State objects as connectors between transmission elements 29 Structure of a scalar variable generalized coordinate 29 Components of a moving frame 240 0200 33 Hierarchy of kinetostatic transmission elements of MIJBILE excerpt 36 Model of a kinetostatic transmission element oaoa a a 38 Terms and computational steps involved in the transmission of motion and TOR COS a an eh e rA eB hee Staton Ry AE ia Bick rake che Bee ER r e Pe Sie Geek 40 Modelofta Teid Hk tani gesi oti be ie ath tie a ra ath Be ad ah Ea ia 44 Model of an elementary joint 02 02 0002 eee 47 A Simple Manipulator a7 ee aaa ee ae EG eae eG 50 Elementary force element attached to a scalar measurement object 52 Model of a mass element o o we oe wie ee eee 4 we 53 Modeling of the inverse dynamics of a SCARA robot 55 Comparision of tree type and closed loop systems 57 Three basic methods for modeling loops in MIJBILE 59 Analysis of a shaker mechanism a 00 eee eee ee 61 Example of a chord aie en Ao os ae Bee Sal eh e te ee BO Ge akg 65 Example of a measurement for a moving object 67 A spatial measurement g 2 4 oe as FAK GS te
2. MoAngularVariable theta_1 theta_2 theta_3 theta_4 joints MoElementaryJoint MoElementaryJoint MoElementaryJoint MoElementaryJoint links connecting joints MoVector r 1 s MoRotationMatrix A R1 K1 R2 KO R3 K4 R4 K6 gt d gt MoRigidLink 1_1k KO MoRigidLink r_lk K2 MoRigidLink s_lk K5 MoRigidLink d_1lk k7 Ki K3 K6 K8 K2 K4 K5 K7 aan R e subsystem to be iterated while MoMapChain coupler 2 coupler lt lt s_lk lt lt R4 lt lt d_lk constraints MoChordPointPointQuadratic core MoChordPointPlane compl_1 MUIBILE 1 3 User s Guide theta_1 theta_2 theta_3 theta_4 ws N N gt gt gt gt yAxis zAxis xAxis xAxis ws ys N solving for constraints K8 coupler K3 KO DO_BRANCH_III DO_BRANCH_II DO_BRANCH_I DO_BRANCH_II K4 core R2 xAxis DO_BRANCH_I gt upper upper lower lower where where input new lower left chain left right unknown force right same as in core lower chain normal of the plane where unknown 91 DO_BRANCH_II DO_BRANCH_III where force DO_BRANCH_III input MoChordPointPlane compl_2a K5 new lower right core same as in compl_1 R3 lower chain yAxis normal of the pla
3. Types of motion and force invocation 2 0 0 0 0 000 2 eae Meaning of the terms in Fig 4 3 02 0 202004 Possible values for the motion subtask selection parameter Possible values for the force subtask selection parameter Geometric types of measurement objects 0 0004 vi 5 2 Basic formulas for spatial measurements kinematics 70 5 3 Basic formulas for spatial measurements statics 0 71 5 4 Elements of the array state for spatial measurements 71 5 5 Basic geometric types of scalar measurements 73 5 6 Types of measurements involving different numbers of frames 75 5 7 Optional parameters for performance optimization of measurement objects 78 7 1 Default rendering geometry for the basic MIJBILE objects 105 7 2 Importing Inventor files into a M JBILE model 2 107 MUIBILE 1 3 User s Guide vii viii MUJBILE 1 3 User s Guide 1 Preface 1 1 What is MOBILE MU BILE is an object oriented programming package designed for the modeling of multi body systems Its main features are e Intuitive representation of mechanical entities as objects capable of transmitting motion and force across the system e Direct modeling of mechanical systems as executable programs allowing the user to imbed the resulting modules in exisisting libraries e Open building block system design making i
4. MUIBILE 1 3 User s Guide 55 Section 4 5 Modeling of a SCARA robot MoFrame KO K1 K2 K3 K4 K5 MoAngularVariable beta_1 beta_2 beta_3 MoElementaryJoint Ri KO Ki beta_1 zAxis MoElementaryJoint R2 K2 K3 beta_2 zAxis MoElementaryJoint R3 K4 K5 beta_3 zAxis MoVector 11 12 pi p2 p3 11 12 pi p2 p3 MoNullState MoRigidLink link_1 Ki K2 11 MoRigidLink link_2 K3 K4 12 MoReal m 2 4 MoInertiaTensor theta MoInertiaTensor 0 021 0 004 0 020 MoMassElement Mi K1 MoMassElement M2 K3 MoMassElement M3 K5 MoMapChain system m theta pl m theta p2 m theta p3 system lt lt Ri lt lt link_1 lt lt R2 lt lt link_2 lt lt R3 lt lt M1 lt lt M2 lt lt M3 l11 y 12 y 1 0 pi y p2 y p3 y 0 5 beta_l q 0 00 beta_2 q 0 20 beta_3 q 0 30 beta_l qd 0 01 beta_2 qd 1 00 beta_3 qd 0 11 beta_l qdd 0 20 beta_2 qdd 0 00 beta_3 qdd 0 25 system doMotion DO_ALL system doForce DO_ALL cout lt lt beta_1 Q cout lt lt beta_2 Q cout lt lt beta_3 Q 56 lt lt beta_1 Q lt lt endl lt lt beta_2 Q lt lt endl lt lt beta_3 Q lt lt endl MUJBILE 1 3 User s Guide 5 Objects for Closure of Loops Multibody systems can feature two fundamental types of structure i tree type structure or ii single or multiple
5. MoAngularVariable phi MoVector 1 MoElementaryJoint R KO K1 phi MoRigidLink rod Ki K2 1 X3 MoReal m MoMassElement Tip K2 m MoReal k c MoLinearSpringDamper springDamper R k c MoMapChain Pendulum Pendulum lt lt R lt lt rod lt lt springDamper lt lt Tip dynami static vars lt static c equation MoVariableList vars lt phi MoMechanicalSystem Dynamics vars Pendulum KO yAxis MUIBILE 1 3 User s Guide Section 8 3 MI BILE for PC with OpenGL 1 MoVector 0 1 0 m 15 phi q phi qd 0 Pendulum doMotion numerical integrator static MoAdamsIntegrator dynamicMotion Dynamics static MoReal dT 0 1 static MoReal tol 0 01 dynamicMotion setTimeInterval dT dynamicMotion setRelativeTolerance tol animation static MoScene Scene Pendulum interface for 3D rendering Scene makeShape KO create shape for inertial frame Scene makeShape R rod create shape for rigid link Scene makeShape K2 SPHERE 0 1 attach ball at end of rod Scene addAnimation0Object dynamicMotion Scene setAnimationIncrement 0 1 Scene Pendulum Parameters 10 mass 30 stiffness 10 damping static MoWidget parameters parameters addSlider m parameters addSlider k parameters addSlider c ooon static MoWidget initCond Scene Pendulum Initial c
6. MoVector Delta_r 2 MoFrameList K_Outs K_Outs lt lt K_Outputi lt lt K_Output2 MoRigidLink binary K_Inputi K_Outs Delta_r Delta_R Delta_R 0 MoZRotation 45 0 DEG_TO_RAD Delta_R 0 MoZRotation 45 0 DEG_TO_RAD Delta_r 0 1 sqrt 2 MoVector 1 1 0 Delta_r 1 1 sqrt 2 MoVector 1 1 0 MUIBILE 1 3 User s Guide 45 Section 4 3 Joints Links and Chains Theoretical background Rigid Link The transmission functions for the rigid link are defined as quantities of frame K are marked with a prime Position forwards R R AR 4 5 r ART r Ar 22 Velocity forwards w ART 0 j Ar ART ART v v a 4 6 Acceleration forwards B re Eor ART zel 4 7 Go wx Wo x Ar Force backwards r AR ARAr T p E e Le as 4 3 2 The Object MoElementaryJoint Elementary transformations i e rotations about a coordinate axis or translations along a coordinate axis play a fundamental role in the modeling of mechanical systems In M BILE such elementary transformations are termed elementary joints and are re alized by objects of type MoElementaryJoint They form the basis for revolute and prismatic joints Combinations of both are not regarded as elementary joints In stead two additional types of joints are introduced for this purpose These are the MoElementaryScrewJoint for a combination of t
7. frames at the endpoints of transformations MoAngularVariable phi angular variable describing rotation MoVector 1 vector for displacement within the link MoElementaryJoint R KO Ki phi object modeling the revolute joint MoRigidLink rod K1 K2 1 object modeling the rigid link MoReal m scalar mass value MoMassElement Tip K2 m generates a point mass attached to K2 MoMapChain Pendulum this object holds the concatenation Pendulum lt lt R lt lt rod lt lt Tip of the previously defined elements The basic constituents of this program are the objects R rod and Tip of type MoElementaryJoint MoRigidLink and MoMassElement respectively The arguments passed to these objects correspond to their inputs and outputs For example KO and K1 are the input and output frames of the revolute joint R respectively and phi is the cor responding rotation variable among others the angle as explained below Accordingly K1 and K2 are the input and output frames of the rigid connection rod while 1 is the corresponding vector separating them The mass element is modeled by a scalar value representing the mass attached to reference frame K2 The three pieces are assembled as a composite system termed Pendulum by making use of the shift operator lt lt Note that the name of the objects in this program is immaterial Also the sequence of definition of the object
8. A second method of computing the entries of the Jacobians is to set at the output of the transmission element all force components equal to zero besides the jth one which is set to Q 1 Transmission of forces then yields at the input of the transmission element a force vector which is identical to the jth column of the transposed Jacobian thus of the jth row 0 otherwise jth an S epee E 4 4 Elimination of the influence of applied forces is achieved by setting the force transmission subtask selection parameter to DO EXTERNAL The advantage of this method is that it is possible to determine only selected rows of the Jacobian and thus the derivatives of only some specific output variables with respect to all input variables This situation typically arises when long chains of transmission elements undergo only a few constraint conditions Note that the procedures described above are computationally not very efficient but due to their simplicity they are well suited for rapid draft style modeling of multibody systems More efficient procedures can be stated using sparse matrix techniques 4 3 Basic Transmission Elements Links Joints and Chains This section describes the basic building blocks for the modeling of open kinematic chains These elements are e rigid links e elementary joints e transmission chains e force and mass elements Some theoretical remarks are interspersed with the description These formulaes are o
9. The residuum of the rotational closure is thus approximated by a skew symmetric matrix whose independent elements can be put together in a vector dy The operation vect thus extracts exactly the independent components from the perturbation AR In M BILE this vector is slightly modified in order to cope with rotations near to 180 The thus ensuing components are on pz u 3 trace AR y dpy w 3 trace AR where p om dp u 3 trace AR 0 for dy gt e 1 for dg lt 70 MUJBILE 1 3 User s Guide Section 5 2 Basic Properties of Measurement Objects and e 107 is a predefined constant that can be changed by the user This modified residuum vector is what is stored in the array state of the rotational measurement The equations for the statics of spatial measurements are summarized in Table 5 3 The objects take here a force Af and or a torque Ar related to the measurement and apply them to the frames Kfrom and Kto Note again that the order of attachment frames has influence on the sign of the applied forces and or torques translational part rotational part fro RE Af from Rom AF Tro RL AT Trom R hom AT Table 5 3 Basic formulas for spatial measurements statics The result of the measurement is stored in an array of linear variables This array is explained below Table 5 4 gives an overview of the state subentries of the array addressed by state fo
10. 11 MUJBILE 1 3 User s Guide Section 4 3 Joints Links and Chains Note that MiJBILE performs no sorting of the kinetostatic elements supplied to the MoMapChain Hence the order of elements in the concatenation sequence is significant Each object must be placed at such a position in the chain that any objects providing values for its inputs are located to the left of this object As a rule of thumb objects that correspond to mechanical components arranged along serial branches must be con catenated in exactly the same order as in the real system starting from the inertial system For parallel branches one can mix up the objects from the different branches but it must be ensured that the subset of objects belonging to the same branch ap pear in correct order relative to one another For example linking the objects above as SimplePendulum lt lt L1 lt lt R1 would have resulted in erroneous calculations although no error message would have been issued Objects of type MoMapChain are again instances of kinetostatic transmission elements This means that chains of transmission elements can be used again as elementary trans mission elements in those settings in which this is allowed For example if the previous chain is to be expanded by another pair of a revolute and a prismatic joint one can reuse the previous chain as follows MoFrame K4 K5 MoAngularVariable beta2 MoVector 12 MoElementaryJoint R2 K2 K3 bet
11. Floating point numbers x R Stack of floating point numbers Base class for three dimensional orthogonal matrices Base class for scalar state objects Heterogeneous list of scalar state objects Three dimensional euclidean vector Stack of three dimensional euclidean vector Elementary transformation matrix for rotation about the x axis Elementary transformation matrix for rotation about the y axis Elementary transformation matrix for rotation about the z axis Base class for elementary rotation transformation matrices Table 3 1 Overview of the basic mathematical objects of MIJBILE The above mentioned entities have been endowed with certain functions operators and data structures that make it possible to use them in an intuitive and mathematically fa miliar way When using the operators care must be taken to regard the correct precedence of the latter This precedence of operators is fixed by the C language For convenience the precedence of operators used in MIYBILE is recollected in Table 3 2 Each block of MUIBILE 1 3 User s Guide 19 operators listed within two horizontal lines constitutes a group The precedence of groups of operators is from top to bottom and within each group operators are applied from left to right Most of the operators defined in MUIBILE are in conformance to common usage in C However some of these are applied quite differently For example the operator for the cross product of two vec
12. 2 2 Example Analysis of a Simple Pendulum 2 2 2 1 Dissection and Re Assembly of the System 2 2 2 Calculating Dynamic Properties 2 2 2 3 Automatic Integration of Dynamical Equations 2 9 OUMA be ts Se e Hs SS ae Gee S oe wl Ee g 3 Basic Mathematical Objects 3 1 The Universal Neutral Element MoNullState 3 2 The Basic Scalar Types MoReal and MoAngle 3 3 Vectors and Matrices 2 4 JB Vector ES 4 2 GA ok Oe Eee Be eee ia 3 3 2 Matrices os a ohh aie ee ht ee he de Saad he at 3 4 Kinetostatic State Objects 204 3 4 1 Scalar Kinetostatic State Objects MoStateVariable 3 4 2 Spatial Kinetostatic State Objects MoFrame 4 Basic Kinetostatic Transmission Elements 10 10 12 12 14 16 18 19 21 22 23 23 24 27 28 32 35 il 4 1 Overview of Supplied Kinetostatic Transmission Elements 4 2 Generic Properties of Kinetostatic Transmission Elements 4 2 1 Model of a Kinetostatic Transmission Element 4 2 2 Invoking Motion and Force Transmission 4 2 3 Selection of Motion and Force Transmission Subtasks 4 3 Basic Transmission Elements Links Joints and Chains 4 3 1 The Object MoRigidLink wes 4S ee ee wh eae es 4 3 2 The Object MoElenentary Joint 4 Gok Ee as 4 3 3 The Object MoMapChain 0200 4 3 4 A simple example oaoa
13. MoMapChain MoForceElement MoConstantWrench MoRigidLink MoLinearSpringDamper MoOutputGenerator MoChord MoCompositeJoint MoElementary Joint MoConstraintSolver MoConstraintSolver MoExplicitSolver MoMassElement MoImplicitSolver MoIntegrator MoDriver Figure 4 1 Hierarchy of kinetostatic transmission elements of MIJBILE excerpt Reference Sheets 4 2 Generic Properties of Kinetostatic Transmission Elements The notion of the kinetostatic transmission elements was already introduced in Section 2 This section focuses on the common features of the kinetostatic transmission elements presenting their underlying generic model and shared services 36 MUJBILE 1 3 User s Guide Section 4 2 Generic Properties of Kinetostatic Transmission Elements Class Functionalit class type MoMap generic base class for kinetostatic transmission elements abstract MoElementaryJoint rotational or prismatic joints concrete MoRigidLink rigid link binary or multiple concrete MoElementaryScrewJoint screw joint concrete MoCompositeJoint base class for multi degree of freedom joints abstract MoSphericalJoint spherical joint both Euler or Byrant angles concrete Mo3DTranslationalJoint pure three dimensional translation concrete MoF loatingBodyJoint general spatial motion concrete MoMapChain concatenation of transmission elements concrete MoForceElement the base class for objects modeling applied forces abstract MoConstantWrench spatial force or momen
14. MoRigidLink L2 K3 K4 12 MoMapChain SimpleManipulator SimpleManipulator lt lt R1 lt lt L1 lt lt R2 lt lt L2 initialization of geometrical data 11 MoVector 0 0 12 2 1 MoVector 0 1 0 simulation int nsteps 10 MoReal forceMagnitude 1 0 betal q 90 0 DEG_TO_RAD initial values beta2 q 45 0 DEG_TO_RAD for int i 0 i lt nsteps i MUJBILE 1 3 User s Guide Section 4 4 Force and Mass Elements SimpleManipulator doMotion DO_POSITION SimpleManipulator cleanUpForces K4 f forceMagnitude MoVector 0 0 1 K4 R SimpleManipulator doForce cout lt lt Torque joint 1 lt lt betal Q lt lt nTorque joint 2 lt lt beta2 qQ lt lt n betal q 90 0 DEG_TO_RAD float nsteps beta2 q 180 0 DEG_TO_RAD float nsteps Note that the numerical value for vectors and matrices passed to kinetostatical transmis sion elements can be redefined anywhere in the program The kinetostatical transmission elements retrieve this information during each motion or force traversal anew Note also that the motion traversal is carried out in this example only at the position level Travers ing of the system at velocity or acceleration level is possible by passing as argument to doMotion the value DO_POSITION DO_VELOCITY DO_ACCELERATION 4 4 Force and Mass Elements Forces and mass effects are modeled in MUJBILE as no
15. compiler The character strings following I and L instruct the compiler 4 MUJBILE 1 3 User s Guide Section 1 7 Style and Symbol Conventions where to look for the header files and the libraries of the MIJBILE package Note that we are using here the environment variable MOBILE_HOME_DIR defined above The argu ment lmobile instructs the compiler to load the MIJBILE library libmobile a The argument ofilename instructs the compiler to create an executable program with the name filename containing the model Executing this file will run the model Another technique for compiling MJJBILE models is to use the UNIX tool make Instead of defining compiler invocation parameters anew for each model one can also lay down the compilation rules in a file named Makefile and locate this file in the directory in which the MJJBILE model is placed Compilation is then automatically accomplished by the command make filename Examples of appropriate Makefile settings for M JBILE models can be found in MOBILE_HOME_DIR examples Makefile MOBILE_HOME_DIR examples Inventor Makefile A Makefile template that is suitable for full M JBILE models including Inventor and NAG capabilities is included under MOBILE_HOME_DIR examples Makefile Inventor NAG Template Instructions for generating compiling and executing Mobile models on Windows PC 98 or NT are more elaborate The reader is referred to Chapter 8 for corresponding details 1 7 Style a
16. gt matrix assign a matrix matriz matrix change of sign of a matrix real matrix matrix scale a matrix matriz vector vector multiply matrix times vector vector matrix gt vector multiply vector by transpose of matrix matrix gt matrix transpose a matrix vector gt matrix generate skew symmetric matrix from vector 2 matrix generate dyadic product of two vectors vector vector Table 3 7 General operations for objects of type MoMatrix The two operators in Table 3 7 generating matrices from vectors are defined as Ay 0 a y vector ay e az 0 az az y Agr 0 ax by Ay by Az by Az bz vector vector dy amp by gt Qy by Qy by Qy b az b azbr Qzby Gz Bz Note that the dyadic product operator used here corresponds to the bitwise exclusive OR operator in native C Thus it has a very low precedence over the additive opera tors and the user must enforce by additional levels of parentheses the correct order of evaluation The columns of matrices are accessible for types MoMatrix MoRotationMatrix and Mo InertiaTensor though three members of type vector These members are denoted by e1 e2 and e3 One can access the column vectors of the aforementioned matrices by appending e1 e2 and e3 to their name Another way of accessing the columns of a matrix for the types listed above is to use the parenthetical expression matriz inder Here indez is the index of the column to be
17. of the same type taken for a second frame Kg Both measurements are taken with respect to the inertial frame Objects of this type are useful when one chain is to follow the motion of another For example one can let one robot prescribe the desired height of an object and ask another to achieve this height by an appropriate control The control is then taken over by a constraint solver MUIBILE 1 3 User s Guide 75 Section 5 2 Basic Properties of Measurement Objects Type simple measurement reconfiguring measurement absolute motion Type I f 9 y relative motion Type H f 9 y absolute difference Type III f 9 9g relative difference Type IV f 9 Figure 5 10 Types of measurements based on number of frames 76 MUJBILE 1 3 User s Guide Section 5 2 Basic Properties of Measurement Objects e Relative Difference Type IV This is the most involved topological mea surement type The measurement corresponds to the difference of two relative mea surements one with respect to the upper segment and one with respect to the lower segment As this upper segment is completely isolated from the system the measurement object needs access to the chain describing its kinetostatics Thus the constructor of this type of
18. returned Allowed index values are 1 2 3 Columns are not defined for objects of type MoXRotationMatrix MoYRotationMatrix and MoZRotationMatrix These represent elementary rotations about a coordinate axis in a compact and efficient manner In fact while behaving similarly to objects of class MoRotationMatrix at the global level these objects store internally just the sine and the cosine of the corresponding rotation angle Table 3 8 shows the additional operations supported for inertia tensors Note that the generic operations defined in Table 3 7 also apply to inertia tensors Also note that the product of two inertia tensors is not defined This would not make any sense as the resulting units would not be compatible with any mechanical quantity The special operations supported for rotation matrices are listed in Table 3 9 Note again 26 MUJBILE 1 3 User s Guide Section 3 4 State Objects Operator usage Action inertia inertia inertia add two inertia matrices inertiat inertia inertia add and assign an inertia matrix inertia inertia gt inertia subtarct two inertia matrices inertia inertia gt inertia subtract and assign an inertia matrix inertia MoNullState gt inertia assign zero matrix Table 3 8 Special operations for objects of type MoInertiaTensor that these are in addition to those of Table 3 7 Now the addition of rotation matrices is not supported since it would destroy the property
19. 3 3 3 4 3 5 3 6 3 7 3 8 3 9 3 10 3 11 3 12 4 1 4 2 4 3 4 4 4 5 5 1 Header files for the basic mathematical objects o ooo aa aaa Header files for the basic kinetostatic transmission elements 2 Header files for constraint generation and solution Header files for generation and solution of equations of motion Header files for animation oaoa 0000000 eee eee Container header files n ae ee ye ee a Iconic representation of the elementary objects of MEYBILE Overview of the basic mathematical objects of MIJBILE Precedence of operators in M JBILE 2004 Objects that are not automatically initialized in M BILE Overview of operations for objects of type MoAngle Overview of operations for objects of type MoVector Properties of the different types of three dimensional matrices General operations for objects of type MoMatrix Special operations for objects of type MoInertiaTensor Special operations for objects of type MoRotationMatrix Special operations for objects of type MoXRotationMatrix MoYRotation Matrix and MoZRotationMatrix 0 5 0000004 Subentries for scalar kinetostatic state objects of type MoStateVariable Subentries for kinetostatic state objects of type MoFrame Selection of kinetostatic transmission elements in MEJBILE
20. Fig 4 2 7 input output position qER qg R 1 Sayi i 1 velocity 4ER geR 2 N p rS 7 acceleration R R Q 0 force QER Q ER a transmission behaviour b input and output coordinates Figure 4 2 Model of a kinetostatic transmission element The transmission of motion involves the mapping of the input variables q to a corre sponding set of output variables gq As well the time derivatives and q of the input coordinates i e the velocity and acceleration are mapped to the corresponding time derivatives g and of the output coordinates Apart from the kinematic trans mission functions the kinetostatic transmission element also induces a transmission of forces This transmission is directed in opposite direction to the transmission of motion mapping the forces Q at the kinematic output of the kinetostatic transmission element to corresponding forces Q at the kinematic input The reasoning behind this kind of modeling is explained in the following theoretical background information Theoretical background Structure of the kinetostatic transmission functions The transmission of motion is governed by the following formulas position q q velocity d J acceleration J h where J represents the Jacobian of the transmission element Jo z ERT 4 1 For the transmission of forces one first assumes that the transmission is ideal i e that it neither ge
21. Inventor Graphic Interface For programming and executing a MUJBILE model the following steps have to be per formed 1 Start Microsoft Visual C 2 Select the menu File New In this window choose Mobile Inventor AppWizard and enter a location direc tory and a project name for the model For example in Fig 8 2 the location typed in is c mobile home examples Inventor and the project name is Pendulum MUIBILE 1 3 User s Guide 113 Section 8 2 MUIBILE for PC with Open Inventor 114 Flot Fiss wottepanes Oe Dooumerds ATi COM Anco SS Appice Propel para Dunia Aecmsce Type Wired fence andide ACNE Pernia Dirin aped Pa ea Dai Libes Fi Dabana Proad a ea haie Lina Loran De iucia Addin Wiad Enator n a Entered Tissi Prec wired HAFI E teres Wicd WF bgph aed 2S DED nen E Caa a ioia kacai T did to unani mahine BF Aone Doriga me i MFC ippa bl whit WFC ipp raid je s Wohi raverica Appian Mobae Opes apia Bc Hees ataurrea irad eam Yen Uy Poop i Figure 8 2 Generate a new Project Also make sure Mobile Inventor AppWizard is selected in the Projects win dow After closing this and the following window with OK the Mobile Inventor AppWizard generates a new project including the three new files in our exam ple MoCmpnt cpp MoCmpnt def Pendulum cpp in the Source Files folder of the Workspace window The file Pendulum cpp where the actual mobile model
22. K3 and K8 The three joint variables at the spherical joint are hereby eliminated from the analysis Thus only the rotations theta_2 theta_3 and theta_4 at the joints R2 R3 and R4 see Fig 5 3c remain as dependent variables for this analysis They are collected in the variable list dependentVars Fig 5 3c illustrates the other components employed in the M JBILE model The indepen dent motion of the loop is subsumed in the transmission chain input which contains the joint R1 and the link r 1k This takes care of the independent motion of the right crank The dependent motion is collected in the transmission chain dependentChain The objective of this chain is to produce the motion of the cut frames K3 and K8 as a function of the three dependent variables In the present case only K8 depends on the aforementioned variables Thus only the three revolute joints R2 R3 and R4 as well as the two connecting links s_lk and d_1k need to be assembled in the dependent chain The link 1_1k in the base does not move during motion It needs to be traversed only once during simulation and is thus not included in any of the two transmission chains described above The closure conditions are evaluated by the object sphere of type MoChord3DPosition which measures the difference vector from the origin origin of K3 and to the origin of K8 The solution of the constraint equations is performed by the object Solver which is of type
23. MUJBILE 1 3 User s Guide Section 1 8 Acknowledgements 1 8 Acknowledgements This work was supported by the Laboratory of Mechatronics Fachgebiet Mechatronik at the Gerhard Mercator University of Duisburg The author thanks the head of the Department Prof Dr Ing habil M Hiller for his support during the development of this software Credits are also due to Mr Thorsten Krupp for much of the coding of the package to Mr Christian Schuster for the porting of M JBILE to PCs as well as to Mr Martin Schneider for many valuable suggestions and bug reports MUIBILE 1 3 User s Guide 7 Section 1 8 Acknowledgements module MoConfig h MoVersion h MoReal h MoRealStack h MoAngle h MoNullState h MoAxis h MoStateVariable h MoVariableList h MoVector h MoVectorStack h MoFrame h MoFrameList h MoMatrix h MoInertiaTensor h MoRotationMatrix h MoXYZRotationMatrix MoQutputGenerator h functionality hardware platform and operating system configuration file version number of current MIIBILE installation floating point numbers stacks of floating point numbers basic properties of angles generic zero state objects for selection of axes linear and angular scalar variables lists of variables three dimensional vectors stacks of three dimensional vectors kinetostatic state of a spatial reference frame lists of frames generic three dimensional matrices rigid body inertia matrices orthogonal matrices representing gener
24. Measurements Measurement objects behave in MEYBILE like any other kinetostatic transmission ele ment The kinematic transmission consists in mapping the motion of the frames to the 64 MUJBILE 1 3 User s Guide Section 5 2 Basic Properties of Measurement Objects measurement quantity The force transmission involves the mapping of force components associated with the measurement to the corresponding spatial forces at the frames For illustration of this concept imagine a clothes line spanned between two hooks one at a wall and one at a tree Fig 5 4 If a wind gust makes the tree sway the length of the clothes line as well as its first and second time derivatives will vary The computation of this length variations corresponds to the motion transmission at position velocity and acceleration level respectively Also depending on the stiffness of the line a scalar tension will be induced which produces corresponding spatial forces at the hooks This represents the force transmission length length rate second time derivative of length doMotion 1 lt gt chord spanned between K1 and K2 3 wall tree doForce 1 tension Figure 5 4 Example of a chord The measurement objects of MEJBILE are characterized by three basic attributes A the geometric type determined by the type of geometric elements point plane line or reference system involved in the measu
25. MoImplicitSolver and thus solves the constraint equations iteratively based on a Newton like algorithm The initialization of Solver is accomplished by three argu ments 1 the measurement object sphere whose vanishing signals the closure of the loop 2 the list of variables dependentVars in which the three dependent variables are subsumed and 3 the transmission chain dependentChain which implements the motion of the cut frames as a function of the dependent variables Note that the number of variables in dependentVars coincides with the number of components of the closure condition Note also that chain representing the input motion is not passed to the solver object This is not necessary because the input motion needs to be evaluated only once prior to the loop closure procedure and not multiple times during the solver iterations After solving the constraints the dependent chain is located appropriately by the solver and one can concatenate more elements to the loop In the example below this is done by attaching a mass element CouplerBody to frame K7 of the coupler link d_1k The complete kinetostatics of the loop can be now subsumed as a transmission chain loopKinetostatics comprising the following three blocks which mirror also the rec ommended loop kinetostatics processing structure to be employed with M JBILE 1 the transmission chain input which implements all motions that can be
26. Section 5 2 Basic Properties of Measurement Objects 5 2 6 Constructing Measurements with Different Numbers of Frames The geometrical types of measurements described above can be applied to between one and four reference frames This is yields the topological type of the measurement The basic entities and notations arising in the definiton of the different topological types of measurement shall be first described based on the most general topological type of measurement the type IV measurement depicted in Fig 5 9 For this measurement the loop is cut apart into two segments One of the segments is denoted the lower segment and the other the upper segment the upper segment being the one that is not connected to the inertial frame The measurement consists in taking homolog measurements g and g in the upper and lower segments respectively and subtracting these two values The upper segment is assumed to be modelled by a transmission chain denoted by This chain starts at the upper base frame K and ends at the target frame K s The two cut frames of the lower segment are denoted accordingly as Kg and Kg where Kg matches K and Kp matches K s at the respective cuts The segment cut introduces a total of three branches that are of interest for the subsequent processing of the loop kinematics These branches have the following meaning e Branch I leads to the base frame Kp of the lower segment e Branch II
27. User s Guide Section 3 1 The Universal Neutral Element MoNullState 3 1 The Universal Neutral Element MoNullState The mathematical entities used in MEJBILE stem from quite different algebraic spaces In each of these spaces there exists a unique point which represents some kind of initial or neutral state which is called the neutral point For example the neutral point of a vector space is the origin or zero vector 0 while for transformation matrices it is the identity matrix In MUBILE it is possible to reset any mathematical object to its value at the neutral element by assigning to it the universal neutral element MoNullState The assignment operator returns again a reference to the object MoNul1State so it is possible to concate nate this resetting operation even when the objects at both sides of the equal signs are of different types This allows one to reset a whole bunch of objects in only one line of code as in include lt Mobile MoBase h gt main MoAngle beta MoZRotationMatrix Rot_z beta Rot_z MoNullState Here the angle beta and the matrix Rot_z are simultaneously reset to zero and the identity respectively Due to efficiency issues not all objects are automatically initialized in MIJBILE when they are defined The most volatile of them come into being with arbitrary values Ta ble 3 3 gives an overview of the objects for which no automatic initialization is performed at definit
28. because the transmission operations are coded as virtual functions the user can invoke these functions also by treating pointers to existing kinetostatic transmission elements as instances of the generic transmission element MoMap The following code fragment illustrates this concept Two concrete kinetostatic transmis sion elements are introduced and subsequenty the motion transmission function is invoked once directly for the given objects and once indirectly by treating them as generic trans mission elements and then using the pointer dereferencing mechanism MoAngularVariable beta MoVector 1 MoFrame Ki K2 K3 MoElementaryJoint R K1 K2 beta MoRigidLink L K2 K3 1 direct invocation R doMotion L doMotion indirect invocation MoMap kinetostaticElements 2 kinetostaticElements 0 amp R kinetostaticElements 1 amp L for int i 0 i lt 2 i kinetostaticElements i gt doMotion DO_ALL MUIBILE 1 3 User s Guide 39 Section 4 2 Generic Properties of Kinetostatic Transmission Elements Note that direct invocation leads to more compact code while indirect invocation allows for automatized treatment of larger assemblies through iterative mechanisms In general the user is free to choose the most convenient form for the application at hand function direct invocation indirect invocation motion object doMotion pointer gt doMotion motion object do
29. carried out before solving the constraints 2 the solver element Solver which contains also the depen dent chain that must be moved while solving the constraints and 3 the post solution element CouplerBody in general a transmission chain which subsumes all tasks that 62 MUJBILE 1 3 User s Guide Section 5 1 Basic Methods of Formulating Loop Closure Conditions can be accomplished after solving the loop constraints Note that as the solver invokes internally the kinematics and statics of the dependent chain it is not necessary or even correct to traverse this chain again after processing the constraint solver The ensuing simulation consists of a kinematical and statical traversal of the system In the kinematic part first the input crank is moved then the kinematics of the constraint solver are processed which implies also moving the dependent chain and then the mass element is put in place in the statics part first the force produced by the mass element is applied then the statics of the constraint solver are processed which involves also traversing the dependent chain and finally the statics of the input crank are computed The value of the force state subentry Q of the variable theta_1 then represents the seeked generalized force that has to be applied in order to move the loop as required include lt Mobile MoElementaryJoint h gt include lt Mobile MoRigidLink h gt include lt Mobile
30. distance from a plane to a point MoChordPointPlane h distance from a point to a plane MoChordPointPointLinear h linear distance between two points MoChordPointPointQuadratic h quadratic distance between two points MoChordList h lists of elementary measurements MoConstraintSolver h generic objects for the resolution of constraint equations MoExplicitConstraintSolver h generator of closed form solutions for scalar constraint MoImplicitConstraintSolver h generator of iterative solutions for general constraints Table 1 3 Header files for constraint generation and solution module functionality MoEqmBuilder h generation of mechanical equations of motion MoDynamicSystem h generation of space state form of dynamical equations MoAdamsIntegrator h numerical integration based on Adams Bashford Moulton method MoBDF Integrator h numerical integration based on Gear s BDF method MoExplicitEulerIntegrator h integration based on Euler method MoRungekuttalIntegrator h numerical integration based on 4th order Runge Kutta method MoStaticEquilibriumFinder h computation of stationary point Table 1 4 Header files for generation and solution of equations of motion module functionality Inventor MoScene h viewer and editor window for 3D motion animation Inventor MoWidget h slider and push button widgets Table 1 5 Header files for animation module functionality MoBase h all of Table 1 1 MoBasicKTE h all of Table 1 2 MoConstraints h all of Table 1 3 M
31. edit mode view mode Revolute joint default rendering Rigid link default rendering OY Rots Roty M m Figure 7 2 Overview of the Inventor interface for MIJBILE Fig 7 2 shows the viewer resulting for the code described above One can see the default rendering for the joint and the link The wireframe cage displayed around the joint is the manipulator that pops up when the user hits the joint with a left mouse click while being in edit mode Edit mode is entered by hitting the arrow icon on the right menu bar with a left mouse button click By hitting the cage figure around the joint and moving the mouse with the left button pressed down the user can operate the joint This is called dragging Besides the dragging operations one can perform viewpoint modification operations This is accomplished by entering view mode View mode is chosen by clicking on the hand symbol on the right menu bar By moving the mouse with the left button pressed down the camera is rotated By moving the mouse with the middle button pressed down the window is paned By moving the mouse up and down with the left and middle mouse buttons pressed down the camera is zoomed out and in respectively Other functionalities of the viewer can be inquired by choosing Help from 106 MUIBILE 1 3 User s Guide Section 7 2 Importing Inventor Files the submenu item functions in the popup menu obtained by pressin
32. effects hydraulics and control theory are also under development and will be included in M JBILE 3 x 1 4 Scope and Organization of this Manual This manual describes the basic software implemented in M JBILE 1 3 and its application to multibody systems Specifically the definition use and application of the objects listed above are described at a syntactical level and illustrated by several examples at a tutorial level The manual does not cover details of the language C and of the implementation of the package MIXBILE Readers interested in these topics are recommended to consult the related literature and or the program listings The manual is organized in two parts e Part I of the manual gives an introduction to the objects of MIJBILE describing their functionality and illustrating their use by several examples Moreover some theoretical background information has been inserted for readers interested in the underlying computations of MIXBILE These insertions are not essential for the use of MUJBILE and can be skipped by the casual reader 2 MUJBILE 1 3 User s Guide Section 1 4 Scope and Organization of this Manual e Part II of the manual comprises the so called M JBILE Reference Sheets a collection of detailed syntax description pages for each entity introduced by the MYUBILE package Part I is structured as follows e Chapter 2 gives a short overview of the capabilities of MIJBILE 1 3 e Chapter 3 describes th
33. function mainLoop passes control to the graphics engine allowing the user to interact with the system These two functions must always be called for the viewer in order for the graphics rendering mechanisms to work properly Instead of just creating a shape the user can also produce so called manipulators for user interaction Manipulators provide the capability of user feed back in addition to pure geometric rendering For example with the joint above being defined as a manipulator the user can accomplish feed back by picking the joint with a left mouse click and dragging the mouse with the left button pressed down The joint then moves according to the mouse motion while the rest of the system follows this motion The code fragment described below realizes this kind of manipulator action MoFrame Ki K2 K3 MoAngularVariable theta MoVector 1 MoElementaryJoint joint K1 K2 theta MoRigidLink link K2 K3 1 MUIBILE 1 3 User s Guide 105 Section 7 1 Creating a Graphic Interface MoMapChain system system lt lt joint lt lt link MoScene scene system scene makeManipulator joint scene makeShape joint link scene show scene mainLoop Currently only manipulators for joints can be created Invoking the makeManipulator function for links frames etc will result in an error joint manipulator animation File Edit Viewing Selecthn Editors Manips Lights Anim
34. gen erate the dynamical equations then solve these and finally animated the ensuing motion All of this shall be accomplished by building a hierarchy of objects that provide more and more complex services by delegating sub responsibilities to other already existing objects 2 2 1 Dissection and Re Assembly of the System The regarded system can be interpreted to consist of a massless link which can rotate about a fixed hinge at one end and to which a point mass is attached to the other end see Fig 2 2 Je phi de K1 a R K2 a system structure b iconic model Figure 2 2 Modeling of a simple pendulum In order to model the system it is first necessary to dissect it into simple pieces Such pieces can be those shipped with the Mf 4BILE library or any other object defined by the user The elementary objects of the MIIBILE library are shown in Table 2 1 In the present example the modeling is based on the following building blocks e an elementary rotation about an axis e a translation within a rigid link and e a mass element attached to a particular location of the system 12 MUJBILE 1 3 User s Guide Section 2 2 Example Analysis of a Simple Pendulum MUUIBILE object MoStateVariable MoRigidLink MoElementary Joint revolute Table 2 1 Iconic representation of the elementary objects of MIJBILE The MUIBILE modeling for the system consists in defining and assembling these pieces MoFrame KO Ki K2
35. means of optimizing performance Note that one can combine different subtasks by using the logical OR operator of C MUIBILE 1 3 User s Guide 41 Section 4 2 Generic Properties of Kinetostatic Transmission Elements name of constant DO_NOTHING DO_POSITION DO_TRANSFORMATION DO_TRANSLATION DO_VELOCITY DO_ACCELERATION DO_EULER COMPUTE_CORIOLIS USE_CORIOLIS DO_CORIOLIS DO_ALL value 0x00000 0x00003 0x00001 0x00002 0x00004 0x00038 0x00008 0x00010 0x00020 0x00030 OxFFFFF meaning term void action transmit rotational and translational motion transmit rotational motion only transmit translational motion only transmit velocity compute and transmit all acceleration terms OL OOO compute and transmit Euler acceleration term only compute quadratic acceleration term only no transmission transmit pre computed quadratic accelera tion term no computation compute and transmit quadratic accelera tion terms both of above do all of the actions defined above Table 4 4 Possible values for the motion subtask selection parameter name of constant DO_NOTHING DO_EXTERNAL DO_INTERNAL COMPUTE_INTERNAL USE_INTERNAL DO_INERTIA DO_ALL value 0x00000 0x00001 0x00006 0x00002 0x00004 0x00008 OxFFFFF meaning void action transmit external forces compute and transmit all source forces compute and store source forces no
36. must match exactly the number of scalar measurements contained in the chord or the chord list passed to the solver Other wise the system of equations would be either over or underdetermined and no solution could be determined The number of scalar variables involved in spatial measurement objects can be hereby determined from Table 5 4 5 3 2 Explicit Solvers Explicit solvers are applicable only when a single constraint equation contains only one single unknown The general syntax for the definition of an explicit solver is e MoExplicitConstraintSolver MoChord amp MoLinearVariable amp e MoExplicitConstraintSolver MoChord amp MoAngularVariable amp In both cases a scalar measurement object must be passed that depends only on one unknown namely the dependenzt variable passed as second parameter This measure ment object must be of the self reconfiguring type i e it must comprise the dependent chain mapping the dependent variable to the cut frames of the measurement The type of the second parameter determines which algorithm the constraint solver applies to the resolution of the constraint When two measurements are taken for one unknown it is possible to determine uniquely a solution The corresponding constructor takes on the form 82 MUJBILE 1 3 User s Guide MoExplicitConstraintSolver MoChord amp MoChord amp MoAngularVariable amp This type of solver is only needed for example for determining the
37. of an angle angle add and assign an angle angle add and assign an angle in radians angle subtract and assign an angle angle subtract and assign an angle in radians angle assign zero angle real return value of angle in degrees real return value of angle in radians real return sine of angle no computation real return cosine of angle no compuation bod AR ded Snes eeder Ser ak cal Table 3 4 Overview of operations for objects of type MoAngle beta3 betal beta2 carry out an algebraic operation beta3 10 DEG_TO_RAD this also works cout lt lt beta3 degrees lt lt n print the value of the angle lt lt beta3 sine lt lt n its sine lt lt beta3 cosine lt lt n and its cosine to standard output cout lt lt betal lt lt n prints 0 0 1 to standard output 3 3 Vectors and Matrices The methodologies for the analysis and synthesis of spatial kinematics and dynamics build substantially upon the notions of three dimensional vectors and matrices In M YBILE three dimensional vectors and matrices are thus given a particular attention These ob jects are represented in MIJBILE by the classes MoVector and MoMatrix Operations concerning these classes have been designed such as to provide the user with an intuitive interface and with optimized code that take best advantage of the three dimensional case 3 3 1 Vectors Table 3 5 shows the operatio
38. of orthogonality Operator usage Action rotation rotation rotation multiply two rotation matrices rotation rotation rotation multiply left times right and assign rotation MoNullState gt rotation assign identity matrix rotation normalize gt rotation normalize non orthogonal matrix Table 3 9 Special operations for objects of type MoRotationMatrix The matrix types MoXRotationMatrix MoYRotationMatrix and MoZRotationMatrix support the same operations plus the additional special operations listed in Table 3 10 Note that it is possible to assign a linear or angular scalar to an elementary rotation ma trix This sets the value of the rotation value keeping the structure of the matrix fixed Note also that it is only possible to assign elementary matrices of the same type to each other For example one can not copy an 2 rotation to a y rotation Operator usage Action XYZrotation angle XYZrotation assign angle of rotation XYZrotation real XYZrotation assign angle of rotation in radians XYZrotation XYZrotation t XYZrotation multiply and assign same type matrix XYZrotation MoNullState gt XYZrotation assign identity matrix Table 3 10 Special operations for objects of type MoXRotationMatrix MoYRotation Matrix and MoZRotationMatrix 3 4 Kinetostatic State Objects Kinetostatic state objects represent entities for storing motion and force information at different places in
39. provided with the MUJBILE software Cur rently the graphic interface of MEJBILE comprises only animation capabilities System modeling must be performed in ASCII format using the objects described in the previ ous chapters In particular graphic modeling capabilities are not covered by the present version These capabilities shall be included in future extensions The MUBILE interface for graphic rendering builds substantially on Inventor the object oriented graphics library of Silicon Graphics Similar functionality can be achieved on other machines after installing an Open Inventor compatible library The basic structure of the graphics interface is depicted in Fig 7 1 There are two basic components involved One is the model of the mechanical system which is realized by assembly of the objects described in the previous chapters Here motions forces etc are computed according to the prescribed input values and or user feedback through the interface The second component concerns the geometric representation of the system s parts on the screen The definition of geometric information and its rendering are realized by modules of the Inventor package Here the man machine interface is provided through which the user can interact with the system This involves moving the camera dragging at joints changing material properties starting an animation etc The interaction between MfABILE and Inventor is provided by a two way link In one
40. required when the user wants to optimize computational performance of the model Depending on the type and number of arguments passed the object will perform the activities described below MoEqmBuilder MoVariableList amp q MoMap amp phi The object will generate the equations of motion for the variables contained in q and using the transmission chain phi for establishing the generalized force vectors and the mass matrix Gravity must be modeled through force elements MUIBILE 1 3 User s Guide 95 Section 6 1 Generating equations of motion class MoEqmBuilder MoEqmBuilder MoVariableList amp q MoMapChain amp phiM MoMap amp phi Same as above only that now a list phiM of transmission elements is supplied in which each transmission element corresponds to the subtree which has to be traversed when establishing the corresponding columns of the mass matrix Note that this list should always contain exactly the same number of elements as the variable list q MoEqmBuilder MoVariableListk q MoMap amp phi MoFrame amp K MoAxis zAxis The object will generate the equations of motion for the variables contained in q and using the transmission chain phi for establishing the generalized force vectors and the mass matrix Gravity will be modeled by applying an appropriate acceleration to the reference frame K in the direction of the coordinate axis supplied as fourth parameter This models gravity in opposite direction to t
41. rod lt lt Tip dynamic equation MoVariableList vars vars lt lt phi 16 MUJBILE 1 3 User s Guide Section 2 2 Example Analysis of a Simple Pendulum MoMechanicalSystem Dynamics vars Pendulum KO yAxis 1 MoVector 0 1 0 m i phi q phi qd 0 numerical integrator MoAdamsIntegrator dynamicMotion Dynamics MoReal dT 0 1 MoReal tol 0 01 dynamicMotion setTimeInterval dT dynamicMotion setRelativeTolerance tol animation MoScene Scene Pendulum interface for 3D rendering Scene makeManipulator R create shape for revolute joint Scene makeShape R rod create shape for rigid link Scene addAnimation0bject dynamicMotion Scene setAnimationIncrement 0 0 animate as fast as possible Scene show MoScene mainLoop move the scene Note that the object Dynamics is now of type MoMechanicalSystem This type is related to the type MoEqmBuilder described above only that it maps the underlying dynamical equations to a system of first order differential equations suitable for numerical integration The definition of the graphical objects above employs the building blocks already employed in the previous mechanic modeling The object Scene of type MoScene takes over the responsibility of rendering the animation for the user It is instructed about which parts to render through the member function makeShape In
42. second complementary angle at a hook joint For a description of the usage of the solver objects the reader is referred to the examples of the next section and the introductory example of Section 5 1 1 5 4 Examples Below three examples of loop closure formulation and solution are supplied The examples cover the three basic methods described previously for stating constraints namely C1 body assembly C2 joint assembly and C3 segment assembly Apart from the pure kinematic modeling the programs generate also the dynamical equations and integrate them using the built in numerical integrators The objects for generation and solution of dynamical equations shall be discussed in the next chapter 5 4 1 Body Assembly of a Spatial Four bar Mechanism include lt Mobile MoSphericalJoint h gt include lt Mobile MoRigidLink h gt include lt Mobile MoImplicitConstraintSolver h gt include lt Mobile MoMassElement h gt include lt Mobile MoAdamsIntegrator h gt main reference frames tiiit itiiti MoFrame KO Ki K2 K3 K4 K5 K6 K7 K8 K9 K10 masses MoReal m_d MoInertiaTensor THETA_d state variables angles MoAngularVariable theta_1 theta_2 theta_3 theta_4 theta_5 theta_6 theta_7 declare joints and connect reference frames MUIBILE 1 3 User s Guide 83 R3 R4 coupler S1 l
43. shape for rigid link Scene makeShape K2 MoSphere so attach ball at end of rod Scene addAnimationObject dynamicMotion Scene setAnimationIncrement 0 1 Scene Pendulum Parameters 10 mass 30 stiffness 10 damping static MoWidget parameters parameters addSlider m parameters addSlider k parameters addSlider c oOoOON static MoWidget initCond Scene Pendulum Initial conditions initCond addSlider phi q 360 360 angle initCond addSlider phi qd 10 10 angular vel Scene show parameters show initCond show return MoScene mainLoop 4 Set the paths to the directories of the include header and lib files as follows see Fig 8 3 Select the menu Tools Options Directories Select Include files in the window Show directories for select your MEJBILE home directory in our example c mobile home Select Libraries files in the window Show directories for select your MUIBILE Inventor lib directory in our example c mobile home lib libInventor 5 Compile the program by selecting Build Build Pendulum dll or F7 The compiler generates the file Pendulum d1ll and writes it into the directory Pendulum Debug or Pendulum Release 6 Start the program To do this open the directory mobile home mobile EXE and double click MobileInventor exe In the then appearing wind
44. the core measurement but instead of the previous lower base frame prev K g the new lower base frame Kz This measurement is then employed to determine the new unknwon MUJBILE allows to model this situation by a constructor of type MUIBILE 1 3 User s Guide 79 Section 5 2 Basic Properties of Measurement Objects Figure 5 11 Measurement Object for Complementary Variable e MoChord MoFrame KB MoChord prev MoMap phi where prev is the previous measurement KB is the new lower base frame and phi is the transmission chain connecting the previous lower base frame o1dKB with the new lower base frame KB If only one unknown remains in the chain between Kg and Ki one can repeat two measurements between the four frames depicted in Fig 5 11 but with different geometric elements than the previous one MJYBILE allows the user to model this situation by making use of a constructor of type e MoChord MoChord prev The only case in which such a situation occurs is when the remaining unkown is an angular rotation and the measurements to be performed are of the geometric type point plane Then the two measurements to be taken are the distances of the origin of frame Kr or K s to the two planes parallel to the rotation axis of the joint containing the new unknown The user must take care that the order of this measurements is such that the vector product of the normal
45. type indicated e Optional parameters are enclosed in square brackets as in ArgumentType Op tional parameters can be left out in which case they are given previously defined default values In MJSBILE optional parameters are not used very frequently In stead one will find several definitions of function calls which differ in the type of the arguments This mechanism is known as polymorphism e Alternative choices are characterized by a vertical bar sponding items An example is separating the corre MoElementaryJoint name lt MoFrame gt lt MoFrame gt lt MoLinear Variable gt xAxis yAxis zAxis Here one of the three choices xAxis yAxis zAxis should be typed as the fourth argument of this constructor ec e Variable number of arguments are indicated by ellipses These indicate that the last pattern can be repeated an arbitrary number of times For example a chain of transmission elements might be defined like this MoMapChain name name lt lt mapi lt lt map2 e Overloaded operators functions constructors and member data are indi cated explicitly in the reference sheets using the C syntax for class definition This assumes some familiarity of the reader with C However this knowledge is limited to recognizing the type and number of arguments passed to the functions or included in the data definition Note that typewriter font is always to be used verbatim 6
46. 106 view mode 106 whereForce 78 wherelnput 78 whereUnknwon 78 MUIBILE 1 3 User s Guide 125
47. 2ESESES gt MoMechanicalSystem SystemDynamic q SolveAll KO zAxis MoAdamsIntegrator SystemIntegrator SystemDynamic MoReal dt 0 1 SystemIntegrator setTimeInterval dt geometry r 1 s d d_s MoNullState MoReal alpha DEG_TO_RAD 10 0 MoXRotation X alpha A X l y 0 8 offset at the base r z 0 2 lenght of 1k 1 S Z 0 2 lenght of 1k 2 d y 0 7 lenght of coupler d_s y 0 5 0 7 position of center of mass masses MoVector theta 4 08895833e 2 1 125e 4 4 08895833e 2 m_d 1 0 THETA_d theta initial conditions hennes theta_l q 1 0 DEG_TO_RAD run the simulation for int i 0 i lt 100 SystemIntegrator doMotion MUIBILE 1 3 User s Guide 93 6 Generating and Solving Dynamic Equations The generation and solution of dynamical is realized in M JBILE as a three step process These steps perform the following operations 1 generation of the equations of motion from a kinetostatic transmission chain class MoEqmBuilder 2 transformation of equations of motion into state space form i e a system of ordinary first order differential equations class MoDynamicSystem and classes derived from it e g MoMechanicalSystem 3 numerical integration of the differential equations class MoIntegrator and classes derived from it e g MoExplicitEulerInteg
48. ARA robot e a scalar parameter describing the mass of the body e an inertia tensor describing its moment of inertia as well as e a vector describing the offset of the center of mass of the body with respect to the origin to which the mass is attached By leaving out the inertia argument one can model point masses By leaving out the offset vector one can model bodies whose center of mass is located at the origin of the frame to which the mass is attached Leaving out such an argument is more efficient than passing an identity matrix or a zero vector respectively The following code fragment ilustrates the use of mass elements MoFrame Ki K2 K3 K4 MoReal m MoInertiaTensor J MoVector s MoMassElement PointCentered K1 m case 1 MoMassElement PointExcentric K2 m s case 2 MoMassElement BodyCentered K3 m J case 3 MoMassElement BodyExcentric K4 m J s case 4 The code generates the following four mass elements e case 1 a point mass attached to the origin of K1 e case 2 a point mass attached to K2 with offset vector s e case 3 a rigid body attached to K3 with center of mass coincident with origin of K3 e case 4 a rigid body attached to K4 with center of mass offest by vector s from origin of K4 4 5 Example Modeling of the Inverse Dynamics of a SCARA robot The following section describes the modeling of the dynamics for a simple robotic system The goal is to compute the joint to
49. Andr s Kecskem thy MUBILE Version 1 3 User s Guide Alle Rechte vorbehalten Nachdruck auch auszugsweise verboten Kein Teil dieses Werkes darf ohne schriftliche Einwilligung des Autors in irgendeiner Form auch nicht zum Zwecke der Unterrichtsgestaltung reproduziert oder unter Verwendung elektronischer Systeme vervielfaltigt oder verbreitet werden 1993 1994 1995 1996 1997 1998 1999 Andr s Kecskem thy Institut fiir Mechanik und Getriebelehre Technische Universitat Graz Important Notice BY OPENING THE SEALING OF THE SOFTWARE ENVELOPE OR BY LOAD ING THE SOFTWARE ON YOUR MACHINE WHICHEVER APPLICABLE YOU ACKNOWLEDGE THE LIMITED WARRANTY AND DISCLAIMER SPECIFIED BE LOW AND CONSENT TO ALL THE CONDITIONS MADE THEREIN IF YOU DO NOT FULLY APPROVE THE TERMS UNDER WHICH THIS SOFTWARE IS LI CENSED DO NOT OPEN THE SOFTWARE ENVELOPE AND RETURN IT INTACT TOGETHER WITH THE WRITTEN MATERIAL HANDED OVER TO YOU OR DO NOT LOAD THE SOFTWARE ON YOUR MACHINE AND DELETE ALL ITEMS RELATED TO THIS SOFTWARE FROM YOUR MACHINE FOR A FULL REFUND OF THE PURCHASE PRICE Limited Warranty and Disclaimer YOU ACKNOWLEDGE THAT THE SOFTWARE MAY NOT SATISFY ALL YOUR REQUIREMENTS OR BE FREE FROM DEFECTS BY THE PRESENT LIMITED WARRANTY IT IS WARRANTED THAT THE MAGNETIC MEDIA ON WHICH THE SOFTWARE IS RECORDED IS FREE FROM DEFECTS IN MATERIALS AND WORKMANSHIP UNDER NORMAL USE FOR 90 DAYS FROM PURCHASE HOW EVER THE SOFTWARE AND THE ACCOMPANYI
50. Boy A Re ee Se Bee A ES 69 Basic form of a scalar measurement 2 02 0000 4 72 Geometric entities involved in the measurements between points and planes 73 iv 5 9 Entities of interest for the topological types of measurement 74 5 10 Types of measurements based on number of frames 76 5 11 Measurement Object for Complementary Variable 80 5 12 Modelling of the spatial Four bar mechanism 84 5 13 Modeling of the Dynamics of a Shaker Mechanism 87 5 14 Modelling of the Shaker explicit solution 90 6 1 Model of the inverse dynamics of a multibody system 95 6 2 Modelling of the TriplePendulum 0 101 7 1 Basic structure of the M JBILE Inventor interface 103 7 2 Overview of the Inventor interface for MIJBILE 2 106 7 3 An example of the use of slider widgets 0 108 8 1 Setting a environment variable anda path 113 6 2 Generate a new Project scx seen we a Re ee Re ee 114 8 3 Setting the path to header and library files 117 8 4 Start of a model with Open Inventor 40 118 8 5 Start of a model with OpenGL 402 4udvee ase Oa ane See a EE 120 8 6 Interactive models 4 4 5 4 d eee doh a otk de week Bete Beane 121 MUIBILE 1 3 User s Guide v List of Tables Tal 1 2 1 3 1 4 1 5 1 6 2 1 3 1 3 2
51. Earr ART 0 RE a 4 11 5 Ew w x uO 5 Ea ARTw x sw xu 2 u Force backwards T AR AR Ar f 0 AR T bas Qe ul 0 fi Qs 0 ul In Eq 4 9 Rot u designates the orthogonal matrix corresponding to a rotation about the axis with direction u by an angle Rot u l3 sinO 1 cos0 ti 4 3 3 The Object MoMapChain 4 13 Mechanical systems usually do not consist of only one part but result typically from the assembly of several components In MUIBILE such composite systems are modeled as chains of kinetostatic transmission elements for which a new type MoMapChain is in troduced Because the overall transmission behaviour of a chain of transmission elements just consists of carrying out the transmission operations of the individual transmission elements chains of kinetostatic transmission elements are simply established by concate nation This concatenation is achieved in MUJBILE by the user with the shift operator lt lt which appends the kinetostatic element to the list to the left For example the following code fragment defines the object SimplePendulum as a concatenation of a revolute joint R and a rigid link L MoFrame KO Ki K2 MoAngularVariable betal MoVector 11 MoElementaryJoint R1 KO Ki MoRigidLink Li Ki K2 MoMapChain SimplePendulum SimplePendulum lt lt R1 lt lt L1 48 betal xAxis
52. Force pointer gt doforce Table 4 2 Types of motion and force invocation The two mechanisms of transmission function invocation are summarized in Table 4 2 The ellipsis in the function call represents an optional argument by which the user can se lect only a subset of the operations involved in the computation of an invoked transmission function These subtasks are described in the following section 4 2 3 Selection of Motion and Force Transmission Subtasks When invoking the motion and force transmission functions one can pass an argument of type MoTransmissionSubtask The effect of the transmission subtask selection param eter is to select particular terms which are to be calculated or added to the result during traversal of the transmission function The individual terms and calculation steps arising in the transmission of motion and force are illustrated in Fig 4 3 The meaning of these terms is displayed in Table 4 3 position velocity 1 acceleration d force Figure 4 3 Terms and computational steps involved in the transmission of motion and forces Table 4 4 lists the possible values of the transmission subtask selection parameter for motion transmission and their relationship to the terms identified above In the setting of spatial transmission elements the term corresponds to the trans mission of the reference acceleration together with the relative acceleration due to the second tim
53. Force to the object s name The invocations can be repeated as many times as required without having to look again into the details of the once modeled object Moreover parameters as well as variables of the transmission elements such as 1 and m and phi can be basically treated as symbols i e they can be assigned actual numerical values at arbitrary locations in the program The variable phi is a representative of a special set of objects in MEXBILE termed scalar kinetostatic state objects These objects comprise information about position velocity acceleration and force of a relative displacement The subentries can be addressed by MUIBILE 1 3 User s Guide 15 Section 2 2 Example Analysis of a Simple Pendulum 4 appending q qd qdd and Q to the name of the scalar kinetostatic state object Thus the entry phi q above addresses the position of the variable phi This entry is for rotational variables an angle which differs from the other scalar quantity the real number in that it contains also the sine and the cosine of the angle This angle is first initialized to the value zero and then incremented within the loop in steps of in 18 steps of 10 Note that the object Pendulum does not come into play within the loop anymore This object is now controlled by the object Dynamics The object Dynamics in turn generates the equations of motion of its subordinate objects her
54. JBILE model in the place indicated in the template Again all variables must be declared as static Not declaring all variables as static will result in loss of information and unpredictable errors TT OT rigor rin ONAT emit i a2 M fi zil Pisi Te dedosa mooo bommi ani qJ les etae ERE padma n out C Prpa ira be RR BL E l Pegg hb ea a muy She SL CPanel Weed Sree De CCL Cue Oboe eel Soci Ce CL D Ppap Micro inn SHA TL MCL LICE amides odes wei et Tih ies E Fi Figure 8 3 Setting the path to header and library files MUIBILE 1 3 User s Guide 117 Section 8 3 MI BILE for PC with OpenGL mobile Input Dialog Suchen in E Debug 7 El c ee Dateiname Pendulum dll Dateityp x Cancel choose a Mobile Model dll Figure 8 4 Start of a model with Open Inventor The same program as in section 8 2 yields then here include include include include include include include include lt Mobile Inventor MoMFC h gt lt Mobile MoElementary Joint h gt lt Mobile MoRigidLink h gt lt Mobile MoMassElement h gt lt Mobile MoLinearSpringDamper h gt lt Mobile MoAdamsIntegrator h gt lt Mobile Inventor MoScene h gt lt Mobile Inventor MoWidget h gt _void main insert here your Mobile code definition of mechanical system see previous section 118 static static static static static static static static static static MoFrame KO Ki K2
55. Kinetostatics lt lt input lt lt Solver lt lt CouplerBody geometry and mass properties r gs d d_s MoNullState MoReal alpha DEG_TO_RAD 10 0 MoXRotation X alpha A X l y 0 8 offset at the base r z 0 2 lenght of r_1k S Z 0 2 lenght of s_1lk d y 0 7 lenght of coupler d_s y 0 5 0 7 position of center of mass m_d 1 0 THETA_d MoVector 4 08895833e 2 1 125e 4 4 08895833e 2 initial conditions theta_1 q 1 0 DEG_TO_RAD compute inverse dynamics loopKinetostatics doMotion DO_ALL propagate motion loopKinetostatics doForce DO_ALL propagate forces cout lt lt Computed torque at joint R1 lt lt theta_1 Q 5 2 Measurement Objects Measurement objects are special objects introduced in MUJBILE to map the motion of moving frames to scalar or spatial quantities The measurements are taken either directly between the frames or between pairs of geometric elements such as points planes lines etc attached to these frames In MS4BILE there are several types of measurement objects also termed chords that are a combination of different geometric topological and activity types Below we reproduce the characteristic features these attributes and the basic functioning of the measurement objects The usage of these objects for constraint equation formulation and solution is explained in Section 5 3 5 2 1 Basic Properties of
56. LE 1 3 User s Guide Section 2 2 Example Analysis of a Simple Pendulum include lt Mobile MoBase h gt these are the header files include lt Mobile MoMapChain h gt containing definitions include lt Mobile MoElementaryJoint h gt for the objects include lt Mobile MoRigidLink h gt used below include lt Mobile MoMassElement h gt eo include lt Mobile MoEqmBuilder h gt void main define the mechanical system see previous section MoFrame KO Ki K2 MoAngularVariable phi MoVector 1 MoElementaryJoint R KO Ki phi MoRigidLink rod Ki K2 1 Jo MoReal m MoMassElement Tip K2 m MoMapChain Pendulum Pendulum lt lt R lt lt rod lt lt Tip create a list of generalized coordinates MoVariableList vars vars lt lt phi create an object for generation of the equation of motion MoEqmBuilder Dynamics vars Pendulum KO yAxis set the numerical values for the configuration to solve 1 MoVector 0 1 0 m dys phi q 0 carry out the analysis for int i 0 i lt 18 phi q PI 18 0 Dynamics buildEquations Dynamics printMass Dynamics printForce Note that this is now an executable program The object Dynamics is now capable of generating the equations of motion upon request This occurs in the program by appending buildEquations printMass and print
57. MUJBILE 1 3 User s Guide 3 Basic Mathematical Objects This chapter describes the basic mathematical objects currently shipped the Mi BILE package The objects introduced here cover the typical mathematical entities encountered in the treatment of spatial kinematics and dynamics These are e linear and angular scalars e three dimensional euclidean vectors e transformation matrices e inertia tensors e linear and angular variables and e spatial reference frames Table 3 1 gives an overview of the classes for basic mathematical objects supplied with the MEABILE 1 3 software class Functionality MoAngle MoAngularVariable MoAngularVariableList MoFrame MoFrameList MoInertiaTensor MoLinearVariable MoLinearVariableList MoMatrix MoNullState MoReal MoRealStack MoRotationMatrix MoStateVariable MoVariableList MoVector MoVectorStack MoXRotationMatrix MoYRotationMatrix MoZRotationMatrix MoXYZRotationMatrix Angles representing elements 0 T the one dimensional Torus Scalar state object storing motion and load state of a cyclic variable List of linear variables Spatial state object storing motion and load state of a reference frame List of spatial reference frames Three dimensional inertia tensor Scalar state object storing motion and load state of a linear variable List of angular variables Generic base class for three dimensional matrix Object representing the universal neutral element
58. MoAngularVariable 29 MIABILE description 1 directory structure 4 features of 1 scope of MIXBILE 1 8 2 scope of M BILE 2 2 2 scope of MIXBILE 3 2 2 MOBILE_HOME_DIR MoChord 57 MoChordList 68 MoCylindricalJoint 46 MoElementaryJoint 46 MoElementaryScrewJoint 46 MoExplicitEulerIntegrator 18 MoFrame 32 state subentries 33 MoFrameList 34 MoInertiaTensor operations 26 MoRungeKuttaIntegrator 15 MoLinearVariable 29 MoMap 123 Section 8 4 MABILLE for PC with Graphic User Interface virtual functions 35 MoMapChain 48 sequence of elements in 14 MoMatrix 23 accessing elements 26 MoNullState 21 MoReal 22 MoRigidLink 44 MoRotationMatrix operations 27 MoScene 17 MoSolver 58 MoStateVariable 29 MoAngularList 31 MoVector 23 accessing elements 23 basic operations 23 MoXYZRotationMatrix operations 27 neutral element 21 numerical integrators Adams Bashfort Moulton 17 objects scalar types 22 basic mathematical 19 iconic representation of 13 not initialized by constructor 21 operators precedence of 20 26 parameter passing pass by reference 14 pass by value 14 planes of projection 73 pose 58 precedence of operators 19 20 prerequisites C 1 kinematics and dynamics 1 reference frames 10 rigid link 44 multiple 45 scalar kinetostatic state objects 11 see state kinetostatic objects scalar scalar load 52 124 applied to a chord 52 app
59. MoImplicitConstraintSolver h gt include lt Mobile MoMassElement h gt void main reference frames MoFrame KO Ki K2 K3 K4 K5 K6 K7 K8 state variables MoAngularVariable theta_1 theta_2 theta_3 theta_4 joints MoElementaryJoint Ri Ki K2 theta_1 yAxis MoElementaryJoint R2 KO K4 theta_2 zAxis MoElementaryJoint R3 K4 K5 theta_3 xAxis MoElementaryJoint R4 K6 K7 theta_4 xAxis ws sy as ee links connecting joints MoVectorr 1l1 s dz ds MoRotationMatrix A MoRigidLink 1_1k KO Ki MoRigidLink r_lk K2 K3 MoRigidLink s_lk K5 K6 MoRigidLink d_lk K7 K8 an KH Veuve subsystem containing only input motion MoMapChain input input lt lt R1 lt lt r_lk subsystem to be iterated while solving the constraints MoMapChain dependentChain dependentChain lt lt R2 lt lt R3 lt lt s_lk lt lt R4 lt lt d_lk constraint solving objects MoChord3DPosition sphere K8 K3 MoVariableList dependentVars dependentVars lt lt theta_2 lt lt theta_3 lt lt theta_4 MoImplicitConstraintSolver Solver sphere dependentVars dependentChain MUIBILE 1 3 User s Guide 63 Section 5 2 Basic Properties of Measurement Objects mass element MoReal m_d MoInertiaTensor THETA_d MoMassElement CouplerBody K7 m_d THETA_d d_s complete kinetostatics MoMapChain loopKinetostatics loop
60. NG WRITTEN MATERIALS ARE LICENSED ASIS ALL IMPLIED WARRANTIES AND CONDITIONS IN CLUDING ANY IMPLIED WARRANTY OF MERCHANTIBILITY OR FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED AS TO THE SOFTWARE AND AC COMPANYING WRITTEN MATERIALS AND LIMITED TO 90 DAYS AS TO THE MAGNETIC MEDIA YOUR EXCLUSIVE REMEDY FOR BREACH OF WARRANTY WILL BE THE REPLACEMENT OF THE MAGNETIC MEDIA OR REFUND OF THE PURCHASE PRICE IN NO EVENT WILL ANY OFFICER OR EMPLOYEE OF THE TECHNICAL UNIVERSITY OF GRAZ OR THE DEVELOPER OF MOBILE BE LIABLE TO YOU FOR CONSEQUENTIAL INCIDENTAL OR INDIRECT DAM AGES INCLUDING DAMAGES FOR LOSS OF BUSINESS PROFITS BUSINESS IN TERRUPTION LOSS OF BUSINESS INFORMATION AND THE LIKE WHETHER FORESEEABLE OR UNFORSEEABLE ARISING OUT OF THE USE OR INABIL ITY TO USE THE SOFTWARE OR ACCOMPANYING WRITTEN MATERIALS RE GARDLESS OF THE BASIS OF THE CLAIM AND EVEN IF THE DEVELOPER OF MOBILE OR AN OFFICIAL OR EMPLOYEE OF THE TECHNICAL UNIVERSITY OF GRAZ HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES Contents 1 Preface 1 1 What is NUS oe ieee ae eae Pe a as 1 2 Intended Audience 4 1 3 Scope of MBILE 1 3 5 a wwe Se fo me GV oh Hw A 1 4 Scope and Organization of this Manual 1 5 File Hierarchy of the M JBILE Package 1 6 Compiler ssues eet ete Sed a 1 7 Style and Symbol Conventions a ooo a a 1 8 Acknowledgements ooo aa a 2 Overview 2 1 Structure of MEIBILE aoaaa aaa aa
61. PRISMATIC cout lt lt MoLinearVariable p gt q break case REVOLUTE cout lt lt MoAngularVariable p gt q break It is also possible to access individual entries of variable lists directly through indexing For convenience two styles of indexing are supplied with the MEJBILE software e FORTRAN style indexing using parenthesis and e C style indexing using brackets With FORTRAN style indexing indices run from 1 to the number of entries in the list With C style indexing indices run from 0 to the number of entries in the list minus one For example in the following program fragment each line accesses exactly the same element of varlist varlist 2 gt qd varlist 1 gt qd varlist 2 gt qdd varlist 1 gt qdd varlist 2 gt Q varlist 1 gt Q 1 0 velocity subentries 2 0 acceleration subentries 3 0 force subentries 3 4 2 Spatial Kinetostatic State Objects MoFrame Spatial kinetostatic state objects store the motion and the load of a moving orthogonal frame Spatial kinetostatic state objects are represented in M JBILE by objects of type MoFrame In MIBILE motions of frames are defined with respect to an implicitly assumed inertial reference frame Ko which is at rest see Fig 3 6 This single moving frame is termed below the actual frame while the inertial frame is denoted as the fired frame The motion of the actual frame comprises a rotat
62. This MUJBILE 1 3 User s Guide Section 5 2 Basic Properties of Measurement Objects measure is suited for resolution of angles at a Hooke joint when the second joint is a planar joint note that the distribution of the geometric elements point and plane to the two involved frames is now reversed in comparision to the previous measure class name MoChordPlanePoint Fig 5 8 illustrates the two geometrical types of measurements most used in multibody analysis Kr Kg a point point b point plane Figure 5 8 Geometric entities involved in the measurements between points and planes Table 5 5 summarizes the scalar geometric measurements and the underlying measurement expressions at position level In these expressions rg and rp denote the radius vectors to the origins of the reference frames Kg and Kp as measured from the inertial system respectively and Rg and Rp are the corresponding transformation matrices from the reference frames to the inertial system The vector ug is a unit vector normal to the plane involved in the measurement In MEJBILE only coordinate planes are allowed in measurements Thus unit vectors can have only one of the three values xAxis yAxis and zAxis MoChord geom entity Kp geom entity Kg PointPlane coordinate plane Rerg Rpgrg Rgug PlanePoint coordinate plane Rere Rere Reug Table 5 5 Basic geometric types of scalar measurements MUIBILE 1 3 User s Guide 73
63. This directory is assumed to be in the following c mobile home but can be changed by the user to any other value If using MYIBILE with Open Inventor two environment variables have to be set to specific directories To accomplish this click the Start icon and then traverse the following pop up menues Start Settings Control Panel System Environment Then perfom the following two steps see Fig 8 1 a type MOBILE_INPUTS in the field Variable and c mobile home geom in the field Value then click on Set b click on path in the list User Variables for and type c mobile home lib libInventor in the field Value then click on Set Remark The path variable specifies the location of the file mobileInvDII dll which is part of the interface between MIJBILE and Open Inventor while the environment variable MOBILE_INPUTS specifies the location of standard graphic files of elementary objects like links or joints MUJBILE 1 3 User s Guide Section 8 2 MI BILE for PC with Open Inventor Figure 8 1 Setting a environment variable and a path 3 Copy the files MobileInventor awx MobileOpenGL awx from the directory mobile home pdf to the directory Programme Microsoft Visual Studio Common MSDev98 Template This allows an easy generation of new MYXBILE programs with the Application Wizard of Visual C 8 2 MUIBILE for PC with Open
64. _2 zAxis MoElementaryJoint R3 K4 K5 theta_3 xAxis MoElementaryJoint R4 K6 K7 theta_4 xAxis ws yas N links connecting joints MoVector r 1l1 s d d8 MoRotationMatrix A MoRigidLink 1_1k KO K1 MoRigidLink r_lk K2 K3 MoRigidLink s_lk K5 K6 MoRigidLink d_lk K7 K8 an KH Veuve subsystem to be iterated while solving for constraints MoMapChain DependentChain DependentChain lt lt R2 lt lt R3 lt lt s_lk lt lt R4 lt lt d_lk constraint Ciit iiit MoChord3DPosition sphere K8 K3 implicit solver for constraint equations MoVariableList dependents dependents lt lt theta_2 lt lt theta_3 lt lt theta_4 MoImplicitConstraintSolver Solver sphere dependents DependentChain mass element MoReal m_d MoInertiaTensor THETA_d MoMassElement CouplerBody K7 m_d THETA_d d_s 88 MUJBILE 1 3 User s Guide create a subsystem for applying the input motion MoMapChain input input lt lt 1_1k lt lt R1 lt lt r_lk create a map chain for the complete kinematics of the loop senessssssanssssssssassesssssansssssasssssssasssssssssssss MoMapChain SolveAll SolveAll lt lt input lt lt Solver lt lt CouplerBody list of indepen
65. _HOME_DIR 1ib run time libraries e g libmobile a MOBILE_HOME_DIR bin utility programs MOBILE_HOME_DIR examples examples and test files MOBILE_HOME_DIR Inventor Inventor graphics library The environment variable MOBILE_HOME_DIR should point to the home directory of the MYABILE package This directory is set by the system administrator during installation of MEXBILE A typical value of MOBILE_HOME_DIR is usr people mobile However there might be a different setting on your system Please consult your system manager for obtaining information about the location of the MIJBILE home directory Depending on which shell you are using setting the value of the environment variable MOBILE_HOME_DIR takes on the form assuming the home directory for MTJBILE is usr people mobile Korn shell ksh export MOBILE_HOME_DIR usr people mobile C shell csh setenv MOBILE_HOME_DIR usr people mobile You can check the value of the environment variable MOBILE_HOME_DIR by typing echo MOBILE_HOME_DIR 1 6 Compiler Issues MU BILE is written in standard C Version 2 0 In order to obtain a running program of a MISBILE model one must compile it using the C compiler installed in the system A typical compiler invocation on a UNIX system has the following appearance CC filename C I MOBILE_HOME_DIR L MOBILE_HOME_DIR 1lib lmobile ofilename Here filename C is the program containing the model and CC is the command for invok ing the C
66. a mechanism For the modeling of mechanical systems two types of kinetostatic state objects are required scalar kinetostatic state objects and spatial kinetostatic state objects Spatial kinetostatic state objects embody the junctures between the mechanical components while scalar kinetostatic state objects represent the wires that pass information about desired motion or forces to the joints of the mechanism MUIBILE 1 3 User s Guide 27 Section 3 4 State Objects Kinetostatic state objects enclose information about the motion and load state at a partic ular location of the mechanical system The kinetostatic state consists of four basic items termed kinetostatic state subentries see also Fig 3 3 position velocity acceleration and force kinematics position velocity acceleration Figure 3 3 State subentries of state objects Note thus that the notion of kinetostatic state object differs entirely from the term state variables known from system dynamics In system dynamics state variables are those appearing as first time derivatives in the state space form of the dynamical equations Here we denote as kinetostatic state objects the collection of position velocity acceleration and load information for a scalar or spatial variable in a mechanism State objects are designed as connectors that are placed between the kinetostatic trans mission elements Fig 3 4 By the connector paradigm it is ma
67. a2 xAxis MoRigidLink L2 K3 K4 12 MoMapChain DoublePendulum DoublePendulum lt lt SimplePendulum lt lt R2 lt lt L2 4 3 4 A simple example The foregoing concepts are be illustrated below by the example of two link manipulator displayed in Fig 4 6 The manipulator consists of two revolute joints R and R with non orthogonal non intersecting axes The vertical direction corresponds to the z axis of the inertial reference frame Ko The first joint rotates about the fixed z axis while the second joint is aligned with the x axis of the moving frame The links are assumed to point in direction of the fixed z axis and the moving y axis respectively Both links have a length of 1 The objective is to evaluate the joint torques when a load fp is applied to the tip of the manipulator in direction of the z axis of the frame K4 The corresponding MUABILE model is displayed below include lt Mobile MoElementaryJoint h gt include lt Mobile MoRigidLink h gt include lt Mobile MoMapChain h gt main definition of system topology MoFrame KO Ki K2 K3 K4 MUIBILE 1 3 User s Guide 49 Section 4 3 Joints Links and Chains 50 Figure 4 6 A Simple Manipulator MoAngularVariable betal beta2 MoVector li 12 MoElementaryJoint R1 KO Ki betal zAxis MoElementaryJoint R2 K2 K3 beta2 xAxis MoRigidLink L1 Ki K2 11
68. ae a tad OS ae bo Behe Gee a Re 4 4 Force and Mass Elements se bin ee Senn eo ee eo a ee 4 4 1 Force Elements 2 0 00 2 eee ee ee es 44 2 Wass Elements s seo cacau ratsua ed ee ye Be ee eae RS 4 5 Example Modeling of the Inverse Dynamics of a SCARA robot Objects for Closure of Loops 5 1 Basic Methods for Formulating Loop Closure Conditions 5 1 1 Example Inverse Dynamics of a Spatial Shaker Mechanism 5 2 Measurement Objects neck d ace ao ed ge ae red we pa ae ae 5 2 1 Basic Properties of Measurements 004 5 2 2 Self Reconfiguring Measurements 00 5 2 3 Lists of Measurements 4 4 6 0 ot ede ee oe a Be a ee 5 2 4 Spatial Measurements coe etl ae eee We ee DS 5 2 5 Scalar Measurements 2 2 2 00002 eee eee 5 2 6 Constructing Measurements with Different Numbers of Frames 5 2 7 Optimizing Performance by Specification of Active Branches 5 2 8 Interlinking Measurements 4 5 e4 4 f 44 amp 4 he ee 4 5 3 Objects for Solving Constraints 2 2 200000048 Dioal Miple solvers va awake ye ea ee eR eed Rede ee oh oe 5 3 2 Explicit Solvers e oh ake ba we Kae Mee ae MOE ees MUJBILE 1 3 User s Guide Ol VEATM PLES e t he we ae Ge a oh OS RO Se BS Oe RE Bie eee 5 4 1 Body Assembly of a Spatial Four bar Mechanism 5 4 2 Joint Assembly of a Shaker Mechanism 5 4 3 Segment Assembly of a Sh
69. age with the measurement objects are explained Applications of solver objects to particular loops are displayed in Section 5 4 MUIBILE 1 3 User s Guide 81 Section 5 3 Objects for Solving Constraints 5 3 1 Implicit Solvers The simpler method for resolving constraints is to use implicit or iterative solution schemes In this case one just has to gather a set of measurement objects describing the closure conditions of the loop s and pass it to the solver together with a list of unkown variables and optionally a transmission element The possible initializations of implicit solvers of constraint equations are e MoImplicitConstraintSolver MoChord amp MoVariableListz This case is suitable when the measurement is of self reconfiguring type i e when it contains the dependent chain mapping the values of the dependent variables to the cut frames involved in the measurement Such solvers can be used for closure conditions of type C1 or C2 described above e MoImplicitConstraintSolver MoChord amp MoVariableList amp MoMap amp This case is equivalent to the first one only that now the dependent chain is passed explicitly to the solver e MoImplicitConstraintSolver MoChordList amp MoVariableList amp MoMap amp This case is applicable to sets of constraint equations gathered at several places in the mechanism and which are to be solved simultaneously Note that the number of variables in the variable list
70. aker Mechanism 6 Generating and Solving Dynamic Equations 6 1 The class MoEqmBuilder 9 205 2 hy Glee Oe ere Se Rave oe eee 6 2 Generating Ordinary Differential Equations in State Space Form 6 2 1 The Class MoMechanicalSystem woke Maedoge be eee oe 6 3 Solving the Differential Equations 2 2 020 6 4 Example Dynamics of a Triple Pendulum 7 Graphic Rendering and Animation 7 1 Creating a Graphics Interface e 4 bate aoa be Soe be ene ba eS 2 lmpoertie Inventor Files a co caved bxece vale Algae te ae a th ae a a 7 3 Prescribing Motion by Sliders 22 0 4 2s eee ee a we See 7 4 Realizing Autonomous Animations 0040048 7 5 Further Animation Capabilities 2 200 4 8 MUXBILE for PC Ook installation atai ye ae te ee Bele Seely en OO ee ee a Og 8 2 MOBILE for PC with Open Inventor Graphic Interface 8 3 MYSBILE for PC with OpenGL Graphic Interface 2 2 8 4 MOBILE for PC with Graphic User Interface MUIBILE 1 3 User s Guide 94 94 97 98 99 100 103 103 107 107 110 111 112 112 113 117 120 iii List of Figures 2 1 2 2 2 3 3 1 3 2 3 3 3 4 3 5 3 6 4 1 4 2 4 3 4 4 4 5 4 6 4 7 4 8 4 9 5 1 5 2 5 3 5 4 5 5 5 6 5 7 5 8 Objects in multibody systems aoao a 11 Modeling of a simple pendulum aaa a 12 Graphic representation of the pendulum example
71. al rotations orthogonal matrices representing elementary rotations printing of intermediate values Table 1 1 Header files for the basic mathematical objects module MoRigidLink h MoElementaryJoint h functionality rigid connections between reference frames revolute or prismatic joint aligned with coordinate axis MoElementaryScrewJoint h screw joint aligned with coordinate axis MoCylindricalJoint h cylindrical joint aligned with coordinate axis Mo3DTranslationalJoint h joint realizing general spatial translation MoSphericalJoint h MoF loatingBodyJoint h MoInstantaneousScrew h MoMapChain h MoConstantWrench h MoLinearSpringDamper h MoMassElement h MoConstantStepDriver h joint realizing spherical motion joint realizing general spatial motion computation of instantaneous screws chains of kinetostatic transmission elements force element applying constant force and or moment linear spring damper force element application of inertia properties of rigid body generation of constant velocity motion Table 1 2 Header files for the basic kinetostatic transmission elements MUJBILE 1 3 User s Guide Section 1 8 Acknowledgements module functionality MoChord3D0rientation h relative orientation between two frames MoChord3DPosition h relative vector between two frame origins MoChord3DPose h relative pose between two frames union of both above MoChordPlanePlane h cosine of angle between two planes MoChordPlanePoint h
72. ame definition of transmission chains A mesciesnent simulation manipulatorChain doMotion move the manipulator cameraChain doMotion move the camera simpleChord doMotion make the measurement extraction of coordinates of approach vector approachVector x simpleChord state 0 q approachVector y simpleChord state 1 q approachVector z simpleChord state 2 q MUIBILE 1 3 User s Guide 67 Section 5 2 Basic Properties of Measurement Objects By including the motion of the camera in the measurement a more compact code results MoFrame EEFrame end effector frame MoFrame cameraFrame MoMapChain manipulatorChain robot kinematics MoMapChain cameraChain camera motion MoVector approachVector difference vector definition of measurement object MoChord3DPosition movingChord EEFrame cameraFrame cameraChain measurement manipulatorChain doMotion move manipulator movingChord doMotion make measurement moving internally the camera extraction of global coordinates of approach vector The use of such self reconfiguring measurement objects can be quite useful when modeling of some types of constraint equations and or force elements as discussed further below 5 2 3 Lists of Measurements Like state variables or transmission elements measurement objects can be concatenated into lists If only the kinetostatic transmission properties of the measurement obj
73. an be attached either to a scalar chord or to an elementary joint In the case that the scalar load is applied to a scalar chord the force is applied to two frames that are not necessarily adjacent A scalar chord is a scalar measurement between two frames For example an object of type MoChordPointPointLinear measures the distance between the origins of two frames A scalar load attached to this scalar mea surement applies a force proportional to the value and to the first time derivative of this measurement Fig 4 7 illustrates the resulting force for this example Further types of scalar chords will be discussed in the next chapter g Figure 4 7 Elementary force element attached to a scalar measurement object In the case that the scalar load is applied to a joint the force is applied to the variable describing the relative motion of the joint as well as to the two frames that are adjacent to the joint Only objects of type MoElementaryJoint i e revolute or prismatic joints are allowed for this kind of scalar loads The following code fragment illustrates the use of scalar loads MoReal k stiffness coefficient MoReal c damping coefficient 52 MUJBILE 1 3 User s Guide Section 4 4 Force and Mass Elements chord based spring damper element MoFrame KChordi Kchord2 MoChordPointPointLinear g KChordi KChord2 scalar measurement MoLinearSpringDamper springi g k c spring da
74. and mass effects that act as leaves attached to the kinetostatic skeleton The concatenation of position velocity acceleration and force transmission functions of the global kinematics yields a function which maps the generalized coordinates and their time derivatives to a set of residual generalized forces Q at the input of the global kinematics This function which I represents the inverse dynamics ee of the system has the structure Q 08 4 4 Wt M qt 4 O a 4 WO t 6 1 where we recollects all externally applied forces and M and Q are the generalized mass matrix and the generalized applied forces respectively The residual forces can be used to generate M and Q by the following simplified procedure 94 MUJBILE 1 3 User s Guide Section 6 1 Generating equations of motion class MoEqmBuilder global kinematics ys wee er eet e weg ee kinematic subsystem OMAHA E E E ee Figure 6 1 Model of the inverse dynamics of a multibody system Set at the input of y for the generalized accelerations 0 then in Eq 6 1 the term IO M q vanishes and the residual vector obtained at the input is exactly Q M Eliminate in the calculation of ye the term this is simply done by switching off effects arising from applied and generalized coriolis and centripetal forces and set a single input acceleration g 1 while all others vanish then the resulting for
75. are MoExplicitEulerIntegrator Solves the differential equations by the forward Euler method Apart from the functions described above the following methods and member data are supplied void reset Read the current state of the system into internal storage MoReal StepSize Step size of the method Note that this is not the interval of integration set by setTimeInterval MoAdamsIntegrator Solves the differential equations by the Adams Moulton method Apart from the functions described above the following methods and member data are supplied void setRelativeTolerancere MoReal amp reltol Set the relative tolerance of the integration to reltol MoRungeKuttaIntegrator Solves the differential equations by the Runge Kutta method Apart from the func tions described above the following methods and member data are supplied void setRelativeTolerancere MoReal amp reltol Set the relative tolerance of the integration to reltol Note this object works only with the NAG library installed 6 4 Example Dynamics of a Triple Pendulum include lt Mobile MoElementaryJoint h gt include lt Mobile MoRigidLink h gt include lt Mobile MoMassElement h gt include lt Mobile MoAdamsIntegrator h gt main definition of the system MoFrame KO K1 K2 K3 K4 K5 100 MUJBILE 1 3 User s Guide Section 6 3 Solving the Differential Equations M2 M2 M3 a principle b iconic representation of the mo
76. are the minimum and maximum values allowed when operating the sliders For angles these values are interpreted in degrees The parameter title is a character string to be displayed as the title of the slider When the user actuates the slider the value of the scalar quantity attached to the slider is updated and the doMotion function of the transmission element attached to the slider widget slider widget is invoked Note that one can use any scalar quantity used within kinetostatic transmission elements as the destination of sliders Thus slider widgets make it possible to change parameters online during simulation In order for the slider widgets to be rendered on the screen it is necessary to invoke the show member function explicitly for them This should be done after the show function of the viewer object has been invoked An example of the use of slider widgets is shown in the program reproduced below The corresponding result on the screen is displayed in Fig 7 3 these sliders change system parameters during simulation E a a these sliders actuate viewer window i the system online Figure 7 3 An example of the use of slider widgets 108 MUIBILE 1 3 User s Guide Section 7 3 Prescribing Motion by Sliders include lt Mobile MoElementaryJoint h gt include lt Mobile MoRigidLink h gt include lt Mobile MoMassElement h gt include lt Mobile MoLinearSpringDamper h
77. as points planes and lines M JBILE provides a whole family of classes for mak ing such measurements which are derived from the abstract super ancestor class MoChord The objects instantiated from these classes are again kinetostatic trans MUIBILE 1 3 User s Guide 57 Section 5 1 Basic Methods of Formulating Loop Closure Conditions mission elements i e they can be used as any other kinetostatic transmission ele ment to propagate motions and forces e Ina next stage one or more objects termed solvers are defined that are set to de termine the dependent relative motions within the loop such that the measurements vanish MIYBILE supplies two classes for this purpose which are both derived from the abstract super class MoSolver One solves the constraint equations by itera tive Newton based procedures This is the universal generally applicable method The other takes a scalar equation and solves it in closed form for an unkown joint variable This method only works for special types of measurements and loop archi tectures In both cases the resulting solver objects behave again like kinetostatic transmission elements supplying a motion and force transmission function The choice of optimal closure conditions and solution strategies for a given multibody system is a non trivial task that renders no unique solution Several approaches exist today for this purpose each having its advantages and disadvantages depe
78. ased determination 43 velocity based determination 43 joint cylindric 46 elementary 46 47 prismatic 46 revolute 46 screw 46 joints elementary see elementary joints kinematic inputs 60 kinematic skeleton 51 kinetostatic state subentries 28 kinetostatic state objects 10 27 connector paradigm 28 scalar 28 types of 10 kinetostatic transmission chain 48 kinetostatic transmission element chains of 48 composite 48 concatenation of 48 kinetostatic transmission elements 10 basic class hierarchy 36 chain of 48 generic model 37 generic properties 36 overview of classes 37 linear coordinates 22 linear variables 29 link binary see binary link multiple see multiple link lins rigid see rigid links list of frames 34 lists of chords see chord lists loop constraint processing 58 lower segment 74 Makefile 5 manipulator 105 MUIBILE 1 3 User s Guide mass kinetostatic element 51 mass generalized 14 mass elements 53 matrices 24 basic operations 26 basic properties 25 class hierarchy 24 columns of 26 data structure 25 index expressions 26 measurements absolute difference type 75 absolute motion type 75 characteristic 57 relative difference type 77 relative motion type 75 scalar see scalar measurements spatial see spatial measurements topological types of 74 mechanical components basic operations 10 responsibilities 11 MoAngle 22 operations 28
79. ble list q The builders of dynamic equations are not kinetostatic transmission elements in the sense defined above They thus do not support the transmission functions doMotion and doForce Instead one can invoke the functions described below Normally the user needs not to be concerned with these functions because they are called internally by the objects described in the subsequent sections They are required only the user needs to have direct information about the terms involved in the equations of motion 96 MUJBILE 1 3 User s Guide Section 6 2 Generating Ordinary Differential Equations in State Space Form buildEquations This generates the equations of motion putting the result in internal storage space saveEquations Copies the system matrices generated by buildEquations to the user accessable arrays These arrays are MoReal MassMatrix Symmetric matrix containing the generalized mass of the system All coeffi cients are filled MoReal ForceVector Vector containing the difference of applied and generalized Coriolis forces solveEquations Determines the system accelerations which are in equilibrium with the applied forces and stores the result in the corresponding state subentries of the variables in the variabe list q A call to buildEquations must be performed prior to this invokation printMass Prints the current values of the generalized mass matrix to standard output A call to buildEquations must be perform
80. c squared distance between two points MoChordPointPointLinear linear distance between two points MoChordPointPlane distance from a base point to a target plane MoChord3DPosition difference of two radius vectors MoChord3D0rientation relative transformation between two frames MoChord3DPose difference vector and relative transformation between two frames Table 5 1 Geometric types of measurement objects MYBILE supplies for each measurement object a pointer state Normally the user does not need to be concerned with this state as it is processed automatically when the measurement object is used as constituent for higher level objects such as solvers spring damper elements etc However if required the user can access the internal state by appending state to the name of the object For scalar measurements this pointer references a linear variable containing the actual position velocity acceleration and load of the measure For spatial measurements the pointer addresses an array of linear variables representing the elements of the vectors and or matrices associated with the measurement For example for objects of type MoChord3DPosition the pointer references an array containing the three components of the difference vector between the origins of the two measured frames This is illustrated by a program example in Section 5 2 2 5 2 2 Self Reconfiguring Measurements Measurements normally just take the actual state of the involved f
81. cable to loops featuring a spherical joint the cut is then performed at the spherical joint and the closure condition is produced by an object of type MoChord3DTranslation C3 Segment Assembly Method The loop is dissected at two joints The loop closure condition is formulated by taking characteristic measurements between the end frames of the two resulting segments and setting these measurements equal The characteristic measurements depend on the type of the two cut joints the types of measurements currently supplied with the MUJBILE software are described in Section 5 2 5 This type of assembly is only applicable to systems of constraint equations that are solvable in closed form It is the most involved of all methods described here however its numerical advantages such as computational efficiency and numerical stability make it a good choice for advanced modeling Currently MUBILE is the only multibody package supporting this kind of modeling MUIBILE 1 3 User s Guide 59 Section 5 1 Basic Methods of Formulating Loop Closure Conditions For all three methods the user has to carry out the following steps 1 decide where to cut the loop apart 2 decide which of the joint variable s of the loop are to be treated as dependent variable s and put these together in an object of type MoVariableList in case there are more than one unknowns the other variables and motions are regarded as independent variables or kinematic i
82. ce Q is exactly the vth column of the generalized mass matrix and by repeating this procedure for all columns one obtains the complete mass matrix Note that in branched systems only the subtree starting at the inertial frame and possessing all objects which are successors to qy needs to be calculated Generation of the equations of motion by this approach requires f 1 traversals of the inverse dynamics for one set of equations where f is the degree of freedom of the system The number of evaluations of the inverse dynamics can be reduced when an iterative method is employed for solving the linear system of equations for the accelerations In order to accomplish the task of generating the dynamical equations the object is ini tialized with the list of independent variables for which to generate the dynamic equations and the kinetostatic transmission chain mapping the motion of these variables to the mo tion of the reference frames at which mass or force elements are attached as well as the mass and force elements themselves Apart form this the user can provide information about the direction of action of gravity as well as the reference frame acting as the inertial frame Furthermore one can give a map list containing for each degree of freedom the subtree which contains all components whose motion depends on this degree of freedom as well as the components places between the degree of freedom and the inertial system This last list is only
83. ce for later values storage and retrieval Thus the user can even change these values during simulation However no velocity and acceleration information is regarded for the relative displacement and orientation Thus this technique works only for quasi static variation of parameters The relative rotation is defined as the transformation matrix from components of the output frame to those of the input frame Concerning the relative displacement there is a slight subtlety to be regarded In the form displayed above the vector Delta_r is 44 MUJBILE 1 3 User s Guide Section 4 3 Joints Links and Chains assumed to be decomposed with respect to the output frame However it is also possible to specify the vector with respect to the input frame via the declaration MoRigidLink Link K_Input K_Output Delta_r Delta_R Note that now the order of appearance of Delta_R and Delta_r is reversed If no relative displacement or no relative rotation occurs within the rigid link one can omit the corresponding arguments yielding optimized code for pure fixed point rotation or pure translation For example the following code introduces a pure rotation and a pure translation link respecively MoFrame K_Inputi K_Outputi MoRotationMatrix Delta_R MoRigidLink PureRotation K_Inputi K_Outputi Delta_R MoFrame K_Input2 K_Output2 MoVector Delta_r MoRigidLink PureTranslation K_Input2 K_Output2 Delta_r Note
84. closure conditions arise in a chain forming a closed loop which set of variables to use as independent generalized coordinates of a subsystem etc MUIBILE 1 3 User s Guide 1 Section 1 3 Scope of MIXBILE 1 3 1 3 Scope of MUIBILE 1 3 MYQBILE 1 3 represents the entry level library for the modeling of multibody systems It covers the following topics e Basic mathematical objects and related operators for calculations in spatial dynam ics scalars vectors matrices orthogonal transformations elementary transforma tions inertia tensors e Elementary building blocks for multibody systems reference frames angular and linear variables elementary joints prismatic and revolute rigid links elementary measurements mapping spatial motion to scalar quantities and tuples thereof ob jects for creating composite chains of transmission elements e Elementary force elements spring damper gravitation e Objects for the resolution of constraint equations either in closed form or iteratively e Objects for the generation of the equations of motion e Objects for the numerical integration of the dynamical equations More sophisticated and also more efficient modeling techniques for multibody systems as for example sparse matrix modeling of Jacobians efficient transmission of inertia prop erties etc will be included in the additional package MUJBILE 2 x which is currently under development Further extensions such as elasticity
85. dSlider m O 10 mass parameters addSlider k O 30 stiffness parameters addSlider c O0 10 damping MUIBILE 1 3 User s Guide 109 Section 7 4 Realizing Autonomous Animations MoWidget initCond Scene Pendulum Initial conditions initCond addSlider phi q 360 360 angle initCond addSlider phi qd 10 10 angular vel Scene show parameters show initCond show MoScene mainLoop move the scene 7 4 Realizing Autonomous Animations Besides motion generation by user feed back MI3BILE provides the capability of real izing autonomous motion animation for appropriate objects Objects for autonomous animation are passed to the viewer through the member function addAnimationObject kinetostatic transmission element The object kinetostatic transmission element is an object derived from MoMap and should exhibit a built in mechanism for progressive motion within the doMotion function A typical example is an instance of a class derived from MoIntegrator The user can supply several such objects by repeated invokation of addAnimationObject These objects are then traversed in the sequence in which they were passed to the viewer during animation Animation then consists in invoking the doMotion function for the supplied objects in order of their addition at constant time intervals The real time interval between doMoti
86. de possible to access the information regarding kinematics and statics at any intermediate place of the multibody system In this setting state objects can be viewed as standardized dual ported RAMs allowing access both from the transmission elements and from the user Hence they allow the user to easily exchange or assemble transmission elements by making changes at one place independent of changes at another place Note that the notion of kinetostatic state objects plays a similar role as the concept of nodes in the finite element method There the latter are introduced to define finite elements independently of one another The representation of state subentries differs from one type of kinetostatic state object to the other In the following the characteristics of the state subentries is discussed separately for scalar and spatial kinetostatic state objects 3 4 1 Scalar Kinetostatic State Objects MoStateVariable A scalar kinetostatic state object comprises the information about position velocity acceleration and force related to one generalized coordinate Fig 3 5 This information is pre wired within the various kinds of transmission elements so the user does not need to bother about how to pass the different types of information correctly As discussed in Section 3 2 two types of displacements may arise in kinematics namely angular and linear displacements Accordingly in MUJBILE there exist two types of scalar kine
87. del Figure 6 2 Modelling of the TriplePendulum MoAngularVariable beta_1 beta_2 beta_3 MoElementaryJoint R1 KO Ki beta_1 xAxis MoElementaryJoint R2 K2 K4 beta_2 xAxis MoElementaryJoint R3 K3 K5 beta_3 xAxis MoVector a_1 a_2 a_3 a_4 MoRigidLink arm_a K1 K2 a_1 MoRigidLink arm_b K1 K3 a_2 MoReal mi 1 0 m2 2 0 MoMassElement Mi Ki m1 a_3 MoMassElement M2 K4 m2 a_4 MoMassElement M3 K5 m2 a_4 MoMapChain pendulum pendulum lt lt Ri lt lt arm_a lt lt arm_b lt lt R2 lt lt R3 lt lt M1 lt lt M2 lt lt M3 geometry and masses a_i a_2 a_3 a_4 MoNullState a_i z 0 5 a_2 z 0 5 a_l y 0 5 a 2 y 0 5 a_3 z 1 5 a_4 z 1 0 mi 1 0 m2 2 0 MUIBILE 1 3 User s Guide 101 Section 6 3 Solving the Differential Equations initial conditions beta_1 q DEG_TO_RAD 30 0 beta_2 q DEG_TO_RAD 10 0 beta_3 q DEG_TO_RAD 10 0 dynamics MoVariableList q q lt lt beta_i lt lt beta_2 lt lt beta_3 MoMechanicalSystem SystemDynamic q pendulum KO zAxis MoAdamsIntegrator SystemIntegrator SystemDynamic MoReal dt 0 1 SystemIntegrator setTimeInterval dt for int i 0 i lt 100 SystemIntegrator doMotion 102 MUJBILE 1 3 User s Guide 7 Graphic Rendering and Animation This chapter describes the graphic interface
88. dent coordinates MoVariableList q q lt lt theta_1 generation of equations of motion and integration Ft itiiti MoMechanicalSystem SystemDynamic q SolveAll KO zAxis MoAdamsIntegrator SystemIntegrator SystemDynamic MoReal dt 0 1 SystemIntegrator setTimeInterval dt geometry r 1 s d d_s MoNullState MoReal alpha DEG_TO_RAD 10 0 MoXRotation X alpha A X l y 0 8 offset at the base r z 0 2 lenght of r_lk S Z 0 2 lenght of s_lk d y 0 7 lenght of coupler d_s y 0 5 0 7 position of center of mass masses tt MoVector theta 4 08895833e 2 1 125e 4 4 08895833e 2 m_d 1 0 THETA_d theta initial conditions iiti titititi ttti theta_1 q 1 0 DEG_TO_RAD run the simulation it it ititi tiiit tiit MUIBILE 1 3 User s Guide 89 for int i 0 i lt 100 SystemIntegrator doMotion 5 4 3 Segment Assembly of a Shaker Mechanism Figure 5 14 Modelling of the Shaker explicit solution include lt Mobile MoElementaryJoint h gt include lt Mobile MoRigidLink h gt include lt Mobile MoExplicitConstraintSolver h gt include lt Mobile MoMassElement h gt include lt Mobile MoAdamsIntegrator h gt main 90 MUJBILE 1 3 User s Guide reference frames Et a MoFrame KO Ki state variables
89. direction MIJBILE puts at disposition to the Inventor library the spatial location of frames of interest onto which geometric information is attached In the opposite direction Inventor modules set the values of selected input variables either through sliders or manipulators and trigger the motion and force traversal of the M YBILE model Mobile Inventor graphic objects mechanical model man machine computation of motions and forces 4 interface animation numerical integration doMotion doForce MoStateVariable Figure 7 1 Basic structure of the MIJBILE Inventor interface 7 1 Creating a Graphics Interface In order to produce a graphic interface the user creates an object of type MoScene and passes to it the M BILE model of the system to be animated This object is termed the viewer The MfXBILE model is responsible for carrying out the computations of motions and forces within the system These motions are transmitted to the viewer MUIBILE 1 3 User s Guide 103 Section 7 1 Creating a Graphic Interface through specially defined frames joints and links to which the user attaches geometric information The graphic interface then renders this geometric information by superposing the geometric information to the computed motion of the frames bodies joints etc Moreover the graphic interface allows the user to move the camera introduce light effects manipulate joints start animations
90. e Pendulum In doing this it acts as a responsible but lazy master that invokes the corresponding transmission functions of its subordinates in order to accomplish the overall transmission behaviour for the complete system Such a delegation of responsibility is typical for the object oriented approach M JBILE makes heavy use of responsibility delegation in the modeling of mechanical systems 2 2 3 Automatic Integration of Dynamical Equations Once generated models of mechanical systems can be passed to further objects capable of performing numerical analysis Examples hereof are eigenvalue analysis or the determi nation of the equilibrium configuration of the system Below we reproduce a program in which the equation of motion of the pendulum is numerically integrated and the ensuing motion is animated via a realistic three dimensional graphic model include lt Mobile MoBase h gt include lt Mobile MoMapChain h gt include lt Mobile MoElementaryJoint h gt include lt Mobile MoRigidLink h gt include lt Mobile MoMassElement h gt include lt Mobile MoAdamsIntegrator h gt include lt Mobile Inventor MoScene h gt void main definition of mechanical system see previous section MoFrame KO Ki K2 MoAngularVariable phi MoVector 1 MoElementaryJoint R KO K1 phi MoRigidLink rod Ki K2 1 MoReal m MoMassElement Tip K2 m MoMapChain Pendulum Pendulum lt lt R lt lt
91. e names of the corresponding objects to the makeShape function The default geometric representation of these objects is reproduced in Table 7 1 The default geometry information for the basic MIXBILE objects is defined in the files MoFrameGeom so MoRotationalJoint so MoPrismaticJoint so MoRigidLink so These files are shipped in the subdirectory Inventor examples of the MZJBILE home directory In order to access the default rendering information the user has to copy these files to the working directory from where the MUIBILE program is to be started 104 MUJBILE 1 3 User s Guide Section 7 1 Creating a Graphic Interface invokation rendering makeShape lt frame gt 7 makeShape lt joint gt lt rigid link gt makeShape lt revolute joint gt makeShape lt prismatic joint gt Table 7 1 Default rendering geometry for the basic MEJBILE objects In the code fragment provided above default rendering information is used for the objects joint and link Note that the makeShape invokation for the rigid link involves also a joint as a first argument This information is required for aligning the base of the link which is represented as a fork with the axis of the joint to which the link is attached The actual rendering of the system occurs in the two lines following the makeShape function invokations The function show Q creates a static image of the system The
92. e basic mathematical objects used in conjuction with the modeling of multibody systems e Chapter 4 is concerned with the basic mechanical modeling elements of MIJBILE 1 3 which are termed kinetostatical transmission elements and which constitute the basis for all objects described lateron e Chapter 5 is devoted to the problem of formulating and solving closed loops e Chapter 6 describes the objects for generating and solving the dynamical equations of multibody systems e Chapter 7 gives an overview of the interface of the MIJBILE package for graphic animation In general the material presented in each chapter builds upon the material contained in the previous ones The reader is encouraged to first browse the chapters in the provided order and then to return to individual chapters to work on the details 1 5 File Hierarchy of the M JBILE Package The software of MUIBILE is organized into several modules each module representing a particular group of modeling elements For example there are modules for joints links generators of closed form solutions etc Each module in M JBILE consists of two parts a file defining the interface of the module the so called header file and a file defining the executable portion of the module the so called implementation file Header files have the suffix h while implementation files have the suffix C Depending on which type of license you have purchased you may or may not pos
93. e body C2 cutting the loop at one joint C3 cutting the loop at two joints For Methods C1 and C2 only implicit solvers are suitable For Method C3 there are some cases in which explicit solvers can be applied These cases are characterized by the fact that by taking the measurement all but one of the unknowns of a loop can be eliminated If this is not possible i e if more than one dependent variable remains in the measurement then set of measurements containing the same number of unkowns must be established and an implicit solver has to be used that solves the corresponding equations simultaneously This happens for example when several loops are coupled yielding a set of scalar measurements that form a system of coupled nonlinear constraint equations In this case one collects the corresponding measurement objects in a chord list and solve the complete set of equations by one implicit solver Solvers of constraint equations behave like any kinetostatic transmission elements i e they supply a motion and a force transmission function The motion transmission function consists in establishing and carrying out the motion of the dependent chain such that the loop stays closed The force transmission function involves the computation of the constraint forces within the loops ans their propagation within the dependent chain such that static equilibrium is achieved Below the basic structure of the constraint solvers and their link
94. e derivatives of the relative coordinates This term is denoted the EULER 40 MUJBILE 1 3 User s Guide Section 4 2 Generic Properties of Kinetostatic Transmission Elements term functionality 1 transmission of position rotation translation 2 transmission of velocity no further subtasks 3 complete transmission of acceleration 3a transmission of input acceleration term only 3b computation of quadratic velocity terms only no transmission 3c transmission of quadratic velocity terms only no computation 4 complete transmission of forces 4a transmission of external forces only 4b computation and transmission of internal forces Ac computation of internal forces only no transmission 4d transmission of internal forces only no computation Table 4 3 Meaning of the terms in Fig 4 3 term The term represents the Coriolis centripetal and gyroscopic accelerations stem ming from quadratic velocity expressions This expression is denoted the CORIOLIS term When several kinetostatic transmission elements are concatenated the resulting CORIOLIS terms are added together In this case the notation is a little misleading as the transmission of CORIOLIS terms at the local level involves also the transmission of the reference acceleration stemming from CORIOLIS terms of the predecessor elements However from the global point of view the nomencature is correct as the calculated terms of the accelerations are only those inv
95. e two techniques the user prefers depends on the application A code fragment in which both methods are empoyed is displayed below include lt Mobile MoElementaryJoint h gt main a arrays of frames MoFrame K 10 MoAngularVariable beta MoElementaryJoint joint K 0 K 1 beta access by indices b lists of frames MoFrame Ki K2 MoFrameList frameList frameList lt lt Ki lt lt K2 fill the list MoFrame p while p frameList getNext returns 0 at the end of list cout lt lt p gt R p gt r lt lt n global components of radius vectors Spatial and scalar kinetostatic state objects are crucial for the concatenation of transmis sion elements They can be regarded as the glue that holds the different parts together Moreover kinetostatic state objects provide the interfaces through which the user can access and manipulate the state related data within the mechanical system This gives to the user complete control and insight over the operations performed within the model This will become more clear in the following chapter when the kinetostatical transmission elements are discussed 34 MUJBILE 1 3 User s Guide 4 Basic Kinetostatic Transmission Elements This chapter describes the basic building blocks of M 3BILE for the kinematic static and inverse dynamics modeling of tree type mechanical systems Further objects suitable for the resolution of constrai
96. ects are of interest the concatenation can take place with objects of type MoMapChain However if the list of chords are to be employed as a set of closure conditions to be solved by a constraint solver the list of measurements must be assembled in a special list type termed MoChordList An example of the use of chord lists is MoFrame K1 K2 K3 K4 MoChordPointPointLinear chordi Ki K2 MoChordPointPointQuadratic chord2 K3 K4 MoChordList chords chords lt lt chordi lt lt chord2 A chord list can be used in any setting allowed for a composite chord i e not where only scalar chords are required The sequence of concatenation is hereby immaterial as the solution of a system of equations does not depend on the order in which the equations are supplied 5 2 4 Spatial Measurements Spatial measurements generate vector and matrix quantities describing the relative pose of two reference frames Currently there exist three types of spatial measurement objects in MOBILE 68 MUJBILE 1 3 User s Guide Section 5 2 Basic Properties of Measurement Objects e objects measuring the relative displacement between the origins of two frames class MoChord3DPosition e objects measuring the relative orientation between two frames class MoChord3D0rientation e objects measuring both the relative displacement and relative orientation between two frames class MoChord3DPose Spatial measurement objects are in
97. ed prior to this invokation printForce Prints the current values of the force vector defined above to standard output A call to buildEquations must be performed prior to this invokation printAcceleration Prints the current values of the resolved generalized accelerations to standard out put A call to buildEquations and solveEquations in this order must be per formed prior to this invokation 6 2 Generating Ordinary Differential Equations in State Space Form For the integration of the equations of motion it is required that they are transformed to space state form The state space form of a system of ordinary differential equations takes the form fly t In MUBILE objects that behave like systems of ordinary differential equations in state space form are derived from the class MoDynamicSystem The common property of these classes is the support of the following set of functions MUIBILE 1 3 User s Guide 97 Section 6 2 Generating Ordinary Differential Equations in State Space Form int getOrder Return the number of state space variables included in the dynamic system void giveYd MoReal t MoReal y MoReal yd Evaluate the function f defined above for time t and state y returning the result of the function i e the time derivative y in the array yd void giveActualConditions MoReal y Returns the actual values of the state variables in the array y void setActualConditions MoReal y Sets the act
98. en restart the animation from the position attained after recon figuration Manipulators store also the speed with which the reconfiguring motion was performed This speed is then used by the integrator as an initial condition 7 5 Further Animation Capabilities Further capabilities of the graphics interface are documented in the MJIBILE reference sheets Furthermore several examples have been shipped with the M 4BILE software in the directory mobile home dir Inventor examples and its subdirectories MUIBILE 1 3 User s Guide 111 8 MUUBILE for PC The MU3BILE for PC version enables the user to write and to display mobile models under Windows 98 and NT in the same way as on UNIX environments Moreover in order to make MEIBILE for PC independent of Open Inventor which requires on PC systems licence fees the graphic interface was implemented additionally using only OpenGL which is a royalty free part of Microsoft Windows Finally MZJBILE for PC integrates a graphic user interface for building models interactively 8 1 Installation Installing MZJBILE on a PC is as easy as counting one through three 1 112 Make sure the following products are installed on your PC Microsoft Visual C 6 0 Microsoft FORTRAN Power Station 4 0 or Visual FORTRAN 5 0 or newer version and if the Open Inventor graphic interface is desired additionally TGS Open Inventor 2 5 0 Copy the Mf 4BILE home directory to your hard disk
99. ended to the normal list of arguments e MoChord whereUnknown whereForce whereInput The functionality of these arguments is described in Table 5 7 variable name informs about which branches contain unkowns motion is evaluated and test forces are applied during computation of the Jacobian only at the tips of these branches whereUnknown whereForce for which branches is force traversal required this corresponds to the branches containing unknowns as well as those containing gen eralized coordinates for statics and dynamics calculations wherelInput which branches contain input motions but not unkowns motion is evaluated for these branches at their tips only once at the start of each iteration Table 5 7 Optional parameters for performance optimization of measurement objects The possible values for the optimization parameters are e DO_BRANCH_I do the calculations for branch I e DO_BRANCH_II do the calculations for branch II e DO_BRANCH_IIT do the calculations for branch III e DO_ALL_BRANCHES do the calculations for all branches default If a variable is ommitted it is given the default value DO_LALL_BRANCHES Note that it is not possible to omit an optimization parameter if one further to the right is to be prescribed Note also that by prescribing a non default value the measurement will do less work than if the variable is ommitted However the user s prescriptions are no
100. equation 79 Coriolis acceleration term see quadratic ac celeration term cut 59 cyclic coordinates 22 decomposition of vectors 33 dependent chains 60 direction of transmission 39 directory structure see M JBILE directory structure DO_ACCELERATION 42 DO_ALL 42 DO_EULER 42 DO_EXTERNAL 42 DO_INERTIA 42 DO_INTERNAL 42 DO_NOTHING 42 DO_POSITION 42 122 DO_TRANSFORMATION 42 DO_TRANSLATION 42 DO_VELOCITY 42 examples accessing scalar variables 30 arrays or lists of frames 34 SCARA robot 54 56 shaker mechanism joint assembly 60 simple manipulator 49 51 simple pendulum animation 16 simple pendulum dynamics 12 f simple pendulum kinematics 5 use of MoAngle 22 use of MoNullState 21 use of MoVector 24 force 51 kinetostatic element 51 definition for a moving frame 32 generalized applied 14 generalized Coriolis and centrifugal 14 source 39 frame actual 82 fixed 32 frames list of 34 geometric types of measurement 65 hardware platforms supported Silicon Graphics 17 hardware platforms supported Hewlett Packard 17 header files 3 container files for 9 including 3 overview of existing 8 9 ideal transmission element 88 implementation file 3 implicit constraint solver 82 indexing C style 32 FORTRAN style 32 inertia 51 inertia tensor Section 8 4 MYIBILE for PC with Graphic User Interface components 53 Jacobians evaluation in MIXBILE 1 3 43 force b
101. equations can be decomposed into a cascade of scalar equations each holding exactly one unknown more than its predecessor and each being solvable in closed form for that un known MUYBILE offers a technique for treating such situations explicitly This technique works only for measurements of type IV above The basic idea for solving a cascade of constraint equations is to pass the previous mea surement objects to the measurement object of a newly introduced equation of the cas cade The first equation of this cascade is called the core equation and the rest are termed the complementary equations In progressing from the core equation to the first complementary equation the situation shown in Fig 5 11 will result In this figure it is supposed that a first measurement was performed as the difference of relative motion between the terminal frames K s and kK of the upper segment and the terminal frame Kx and the previous base frame prev Kg of the lower segment It is now assumed that the next unknown to be determined occurs somewhere between the previous base frame prev K z and the new base frame Kp of the lower segment Let denote the transmission chain that connects the previous lower base frame prev Kpg to the new lower base frame Kpg such that it contains exactly one new unkown and that there are no additional unkowns between Kg and K s Then one can take a new measurement which has the same cut frames Kz K and Kp as
102. estandable iterative methods 5 1 Basic Methods for Formulating Loop Closure Conditions The basic procedure for tackling multibody loops in MUBILE is to first dissect the orig inally closed loop into serial chains and then to bring again the loose ends of the serial chains together by requiring the fulfillment of appropriate closure conditions Hereby the following three basic methods are possible see Fig 5 2 C1 Body Assembly Method The loop is dissected at a body The loop closure condition corresponds to equality of pose for the two reference frames at both sides 58 MUJBILE 1 3 User s Guide Section 5 1 Basic Methods of Formulating Loop Closure Conditions gt e ai gt J J ft ft an Ss C1 Body Assembly C2 Joint Assembly C3 Segment Assembly Figure 5 2 Three basic methods for modeling loops in MTJBILE of the cut where the term pose stands for displacement and rotation This kind of assembly is applicable to any type of loop In may however perform only poorly both in terms of computational efficiency and in terms of numerical stability of the computed solution In MUJBILE pose closure conditions are generated by objects of type MoChord3DPose C2 Joint Assembly Method The loop is dissected at a joint The loop closure condition consists in the equality of the geometric elements left invariant by the joint for the two reference frames at both sides of the cut Currently this type of assembly is only appli
103. eta MoElementaryJoint Hinge K_Input K_Output Theta rotational joint Note that the type of joint i e prismatic or revolute is selected automatically according to the type of state variable passed in the first case it was a linear variable while in the second it was an angular one The axis of rotation or translation of elementary joints is implicitly assumed to be the z axis One can overwrite this setting by specifying explicitly the axis through a fourth parameter Possible values for this parameter are xAxis yAxis and zAxis For example a rotational joint R rotating about the y axis and a prismatic joint P translating along the x axis are specified by MoFrame Ki K2 K3 MoAngularVariable Theta MoLinearVariable s MoElementaryJoint R Ki K2 Theta yAxis MoElementaryJoint P K2 K3 s xAxis Theoretical background Transmission functions for elementary joints Below the transmission equations are reproduced for an elementary cylindric joint The equations can be specialized to the case of a revolute or prismatic joint by setting s 0 or 0 respectively The axis is described by a unit vector u u Uy Uz Position forwards R R AR AR Rot u 9 ie r AR r us 49 MUIBILE 1 3 User s Guide 47 Section 4 3 Joints Links and Chains Velocity forwards w ell Ehr ART 0 a 6 4 10 Acceleration forwards w w ART 0 u 0 a Ew ei
104. etc This produces a virtual prototyping environment in which the user can assess the functioning of the system as if it where operating in real world The following code fragment illustrates the use of the graphics interface of MJJBILE MoFrame Ki K2 K3 MoAngularVariable theta MoVector 1 MoElementaryJoint joint K1 K2 theta MoRigidLink link K2 K3 1 yas MoMapChain system system lt lt joint lt lt link MoScene scene system scene makeShape joint scene makeShape joint link scene show scene mainLoop The object scene of type MoScene is the actual viewer of the system This object provides the graphic interface to the user The mechanical system to be rendered is passed to it as an argument The viewer takes control over this system providing data for input motion and invoking motion and force traversals according to the commands of the user At the outset no graphic information is produced by the viewer The MIJBILE model just represents a skeleton within which motion and force are computed but no visible geomet ric properties are considered In order to make parts visible the user invokes the function makeShape of the viewer for each part to be rendered M JBILE provides default geometries for each of the basic modeling objects MoFrame MoElementaryJoint and MoRigidLink In order to use default rendering information the user passes just th
105. g the right mouse button anywhere within the viewer window 7 2 Importing Inventor Files Besides using default rendering geometry MEJBILE can be instructed to employ user defined Inventor files for geometric rendering Inventor files are imported into the M JBILE model by supplying the file name as an argument to the function invokations makeShape or makeManipulator There are three modes for loading Inventor files into the MiABILE model These three modes are recollected in Table 7 2 In this table file represents the name of the file holding the Inventor model for the geometry to be ren dered For information about writing Inventor files please consult the Inventor Mentor manual provided with the Inventor software invokation action attach contents of file to a frame attach contents of file to a revolute joint attach contents of file to a prismatic joint attach contents of file to a revolute joint attach contents of file to a prismatic joint makeShape lt frame gt file makeShape lt revolute joint gt file makeShape lt prismatic joint gt file makeManipulator lt revolute joint gt file makeManipulator lt prismatic joint gt file Table 7 2 Importing Inventor files into a MIJBILE model Besides by direct writing one can also produce Inventor files by translating from other standards to Inventor format For example one can translate AutoCAD dxf file
106. gt include lt Mobile MoAdamsIntegrator h gt include lt Mobile Inventor MoScene h gt include lt Mobile Inventor MoWidget h gt void main definition of mechanical system see previous section MoFrame KO Ki K2 MoAngularVariable phi MoVector 1 MoElementaryJoint R KO Ki phi MoRigidLink rod Ki K2 1 Jos MoReal m MoMassElement Tip K2 m MoReal k c MoLinearSpringDamper springDamper R k c MoMapChain Pendulum Pendulum lt lt R lt lt rod lt lt springDamper lt lt Tip dynamic equation MoVariableList vars vars lt lt phi MoMechanicalSystem Dynamics vars Pendulum KO yAxis a MoVector 0 1 0 m phi q phi qd 0 numerical integrator MoAdamsIntegrator dynamicMotion Dynamics MoReal dT 0 1 MoReal tol 0 01 dynamicMotion setTimeInterval dT dynamicMotion setRelativeTolerance tol animation MoScene Scene Pendulum interface for 3D rendering Scene makeShape KO create shape for inertial frame Scene makeManipulator R create manipulatorfor revolute joint Scene makeShape R rod create shape for rigid link Scene makeShape K2 MoSphere so attach ball at end of rod Scene addAnimation0bject dynamicMotion Scene setAnimationIncrement 0 1 animate in real time if possible MoWidget parameters Scene Pendulum Parameters parameters ad
107. he coordinate axis supplied as fourth parameter If the fourth parameter is ommitted gravity will be applied in negative z direction MoEqmBuilder MoVariableList amp q MoMapChain amp phiM MoMap amp phi MoFrame amp K MoAxis zAxis Same as above only that now a list phiM of transmission elements is supplied in which each transmission element corresponds to the subtree which has to be traversed when establishing the corresponding columns of the mass matrix Note that this list should always contain exactly the same number of elements as the variable list q MoEqmBuilder MoVariableList amp q MoMap amp phi MoFrame amp K MoVector amp gravity The object will generate the equations of motion for the variables contained in q and using the transmission chain phi for establishing the generalized force vectors and the mass matrix Gravity will be modeled by applying an appropriate acceleration in direction of the vector gravity to the reference frame K The magnitude of this vector is ignored MoEqmBuilder MoVariableList amp q MoMapChain amp phiM MoMap amp phi MoFrame amp K MoVector amp gravity Same as above only that now a list phiM of transmission elements is supplied in which each transmission element corresponds to the subtree which has to be tra versed when establishing the corresponding columns of the mass matrix Note that this list should always contain exactly the same number of elements as the varia
108. he reader at this point These classes will be explained in more detail in the subsequent sections At present it is sufficient to grasp the basic ideas behind the implementation of the five steps described above and the use of objects of type MoChord and MoSolver R4 theta_4 R1 theta_1 R2 theta_2 R3 theta_3 a closed system b opened system theta_4 2 CouplerBody R4 pa H d_lk sphere 7 7 K3 G S theta_1 theta_3 theta_2 c iconic model Figure 5 3 Analysis of a shaker mechanism The regarded system involves seven joint variables two at the revolute joints R1 and R4 two at the Hooke joint which is modeled by two revolute joints R2 and R3 with orthogonally intersecting axes and three at the spherical joint sphere see Fig 5 3a According to the six general spatial loop closure conditions only one of these can be chosen as kinematical input which is here the rotation of joint R1 MUIBILE 1 3 User s Guide 61 Section 5 1 Basic Methods of Formulating Loop Closure Conditions For the formulation of the loop kinematics one of the three methods described above has to be selected Because of the occurence of a spherical joint method C2 i e the Joint Assembly Method is appropriate This yields the open structure depicted in Fig 5 3b The cut at the spherical joint introduces three constraint equations corresponding to the concurrence of the origins of the two cut frames
109. home mobile EXE like in section 8 3 the option Interactive see Fig 8 5 brings up a window with a tool bar on the right side and the initial coordinate frame in the center see Fig 8 6 Here mechanical systems can be modelled including the closure of loops Finally the dynamic equations can be solved allowing the animation of the system Furthermore the C code of the model can be exported in the menu File Export Source Code Further details for usage of this module can be recognized from the icons displayed in the decoration of the window 120 MUIBILE 1 3 User s Guide Section 8 4 MABILLE for PC with Graphic User Interface M SceneViewer MobileGL File Edit Insert Extras Animation View KS E Al el S Bereit Figure 8 6 Interactive models MUIBILE 1 3 User s Guide 121 Index C see implementation file h see header files lt lt in kinetostatic transmission chains 48 in variable lists 31 abstract class 35 activity types of measurement 65 angle 16 angular variables 29 animation 110 base frame 72 basic mathematical objects 19 basic mathematical objects 10 categories of objects 10 chord 64 chords 57 client server paradigm 11 closed loop systems 57 closure conditions 57 compiler invocation 4 complementary equations 79 COMPUTE_CORIOLIS 42 COMPUTE_INTERNAL 42 concrete class 35 connector paradigm 29 constraint equations 57 core
110. ide Section 3 3 Vectors and Matrices manner than that of the other classes Note that not all operations allowed for one type apply also to the other MoRotationMatrix MoInertiaTensor MoXRotationMatrix MoYRotationMatrix MoZRotationMatrix Figure 3 2 Class hierarchy for the different types of matrices in M YBILE Table 3 6 displays the basic data structure of the different types of matrices supported by the MUIBILE package Furthermore the 3 x 3 zero matrix MoNul1Matrix and the 3 x 3 identity matrix MoIdentityMatrix are defined class MoMatrix MoRotationMatrix MoXRotationMatrix MoYRotationMatrix MoZRotationMatrix MoInertiaTensor type of space A R x R A SO 3 A E Rot z 0 A E Rot y 6 A E Rot z 6 structure of matrix a11 Q12 443 A ag G22 az Q31 432 433 ATA l 1 0 0 A 0 cos sin 0 sinf cos cos 0 sind A 0 1 0 sinO 0 cos cos sin d 0 A sind cosd 0 0 0 1 ail 412 413 AE A A gt 0 A ayn az az Q13 422 433 Table 3 6 Properties of the different types of three dimensional matrices The generic set of operations for matrices is summarized in Table 3 7 These operations are defined for all types of matrices Specialized operations which only make sense for a particular type of matrix are discussed further below MUIBILE 1 3 User s Guide 25 Section 3 3 Vectors and Matrices Operator usage Action matriz matrit
111. include lt Mobile MoLinearSpringDamper h gt include lt Mobile MoAdamsIntegrator h gt include lt Mobile Inventor MoScene h gt include lt Mobile Inventor MoWidget h gt _void main insert here your Mobile code definition of mechanical system see previous section static MoFrame KO Ki K2 static MoAngularVariable phi static MoVector 1 static MoElementaryJoint R KO K1 phi static MoRigidLink rod Ki K2 1 LS static MoReal m static MoMassElement Tip K2 m static MoReal k c static MoLinearSpringDamper springDamper R k c static MoMapChain Pendulum Pendulum lt lt R lt lt rod lt lt springDamper lt lt Tip dynamic equation static MoVariableList vars vars lt lt phi static MoMechanicalSystem Dynamics vars Pendulum KO yAxis 1 MoVector 0 1 0 m 1 phi q phi qd O Pendulum doMotion numerical integrator static MoAdamsIntegrator dynamicMotion Dynamics static MoReal dT 0 1 static MoReal tol 0 01 dynamicMotion setTimeInterval dT dynamicMotion setRelativeTolerance tol animation MUIBILE 1 3 User s Guide 115 Section 8 2 MUIBILE for PC with Open Inventor static MoScene Scene Pendulum interface for 3D rendering Scene makeShape KO create shape for inertial frame Scene makeManipulator R create manipulatorfor revolute joint Scene makeShape R rod create
112. ink_1 link_2 base R2 a closed system K6 theta_4 R4 core K10 oe K5 M coupler_b K9 pa theta_5 K4 K8 theta_6 theta_7 link link_2 K1 theta_1 I o K3 R1 T R2 theta_2 K0 BE base K2 c iconic representation of the model Figure 5 12 Modelling of the spatial Four bar mechanism MoElementaryJoint R1 KO K1 theta_1 xAxis MoElementaryJoint R2 K2 K3 theta_2 xAxis MoElementaryJoint R3 K4 K5 theta_3 zAxis MoElementaryJoint R4 K5 K6 theta_4 xAxis MoVariableList CardanAngles CardanAngles lt lt theta_5 lt lt theta_6 lt lt theta_7 MoSphericalJoint Si K8 K9 CardanAngles BRYANT_ANGLES link transformations 84 MUJBILE 1 3 User s Guide MoVector r 1 s d_a d_b r 1 s d_a d_b MoNullState MoRotationMatrix A A MoNullState declare links for rigid body transformations ninnninininnnninininninnimnninimmninmnninimnnimmnimmnininimninin MoRigidLink base KO K2 1 A MoRigidLink link_1 Ki K4 1r MoRigidLink link_2 K3 K8 s MoRigidLink coupler_a K6 K7 d_a MoRigidLink coupler_b K9 K10 d_b subsystems for solution of constraints SfefSfSfS2S2S2S2S2S2S2S2222S222S2S22222E2222252E22222EES2 MoMapChain DependentChain Dp2 RightBranch RightBranch lt lt Ri lt lt link_i DependentChain lt lt R2 lt lt link_2 lt lt R3 lt lt R4 lt lt coupler_a
113. ion time The table also exhibits the values that the objects will take upon as class value generated by MoNul1State MoAngle zero degrees MoInertiaTensor MoZeroMatrix MoRotationMatrix MoIdentityMatrix MoVector MoNullvector MoXRotationMatrix MoIdentityMatrix MoYRotationMatrix MoIdentityMatrix MoZRotationMatrix MoIdentityMatrix Table 3 3 Objects that are not automatically initialized in MIJBILE signment of the universal neutral element All other objects of MIJBILE are initialized to a definite value at definition time The corresponding initialization values are described in the MUIBILE reference sheets MUIBILE 1 3 User s Guide 21 Section 3 2 The scalar entities MoReal and MoAngle 3 2 The Basic Scalar Types MoReal and MoAngle Scalar algebraic objects constitute the basis for the contruction of mathematical expres sions and of more involved composite elements In the analysis of mechanical systems there are two types of scalar numbers Fig 3 1 a linear coordinates which are elements of the real line z R and b cyclic coordinates which are elements on the real circle 6 T In MUIBILE these two kinds of scalar algebraic objects are termed MoReal and MoAngle The distinction between linear and cyclic scalar coordinates is appropriate for two reasons e by introducing cyclic coordinates one can avoid repeated evaluation of trigonometric expressions thus reducing computational overhead e b
114. ional part and a translational part The translational part is determined by the radius vector r connecting the origin of the fixed frame with the origin of the actual frame as well as the velocity v and the acceleration a of the origin of the moving frame with respect to the fixed frame The rotational part consists of the orthogonal matrix R representing the transformation of coordinates from the moving frame to the fixed frame as well as the angular velocity w and angular acceleration w of the moving frame with respect to the fixed frame The load at the frame is described by a moment 7 measured with respect to the origin of the actual frame and a force f both of which are assumed to be acting from the actual frame onto the structure spanned between the actual frame and the fixed frame 32 MUJBILE 1 3 User s Guide Section 3 4 State Objects Ie I rotation translation angular velocity linear velocity angular acceleration linear acceleration torque force E Ko K m4 Me lt CB DW Figure 3 6 Components of a moving frame In MY3BILE all vectors are usually assumed to be decomposed in the actual frame Exceptions from this rule are stated explicitly The state subentries for spatial reference frames are accessed by appending the name of the correspondent member element to the name of the frame The identifiers for the member elements are recollected in Table 3 12 state subentry access of type of entry compo
115. is vector behaves like a vector of angular velocity as explained below MUIBILE 1 3 User s Guide 69 Section 5 2 Basic Properties of Measurement Objects translational part rotational part Ar Roo rro Rerom lrrom vect AR vect Rivom Ri Av Rro VTo 7 Rerom VFrom Aw Rro WTo Rerom WFrom Aa Ri aTo R From aFrom Aw Rr WT E R From WFrom Table 5 2 Basic formulas for spatial measurements kinematics An infinitesimally small rotation R can be written as 0 dp dy R Is Oy 0 dpy kbt dy dyy dz 0 M symmetric skew symmetric Thus an infinitesimal rotation can be decomposed in the sum of the identity transformation and a perturbing part that is purely skew symmetric Hereby the skew symmetric part has three indepen dent elements that can be put together in a vector 69 Yz Yy Yz The original skew symmetric matrix is then reconstructed by making use of the tilde operator Regarding now a general matrix AR one can decompose 1 1 AR 5 AR ART 5 AR ART eo seo symmetric skew symmetric If AR is only a small deviation from the identity transformation as expected for example in a Newton like procedure its symmetric part is approximated by the identity matrix and its skew symmetric part is approximated by a skew symmetric perturbation term as described above Hence the closure condition takes the form g AR AR l AR ART 0 NI
116. itialized with two frames the from frame which acts as the base for the measurement and the to frame which is the target The discerning of these two frames is significant only when the sign of a scalar measurement or the tensorial quantities involved in a spatial measurement are of interest If the user is only interested in establishing a measurement for later use in a solver the order is immaterial An example of an initialization of spatial measurements is MoFrame Kfromi Ktoi MoChord3DPosition distance Ktol Kfromi MoFrame Kfrom2 Kto2 MoChord3DPose pose Kto2 Kfrom2 Theoretical background Kinetostatics of Spatial Measurements Objects for spatial measurements give rise to two geometric entities related to rigid body motion the vector Ar pointing from the origin of the from frame Kgom to the origin of the to frame Kito and the transformation matrix AR transforming from components with respect to Kto to components with respect to Kom Fig 5 6 AR K to oo OR K from Figure 5 6 A spatial measurement The formulas involved in the kinematics of the spatial measurements are summarized in Table 5 2 The operator vect in the rotational part extracts three independent quantities from the rotation matrix that are used for formulation of the rotational constraint equations These three numbers can be regarded as coefficients of a vector representing a small rotation increment Th
117. ized joint that supports a doMotion function For this reason integrators have been defined in M XBILE as derivations of the class MoMap for general kinetostatic transmission elements The base class for objects for integrating dynamical equations is MoIntegrator Inte grator objects are constructed simply by passing a dynamic system whose differential equations are to be integrated The constructor is simply MoIntegrator MoDynamicSystem amp sys The object is then capable of solving the differential equations established by sys The currently implemented methods for integrator objects are void doMotion Progresses along the solution trajectory by the time interval specified by the user via the function setTimeInterval Note that this time interval is not necessarily the step size of the integrator but the period after which a result is returned void doForce Void function does nothing int getNumber0fSteps Return current number of already performed steps void setTimeInterval MoReal amp dt Set the time interval for the method doMotion described above to dt MUIBILE 1 3 User s Guide 99 Section 6 3 Solving the Differential Equations void setStartTime MoReal amp t Set the start time of the integration to t Particular implementations of integrators may make it necessary to extend these methods by further functions Currently the following integrator schemes are incorporated into the MIIBILE softw
118. l objects in the sense that they participate in the power transmission of the system Nevertheless they are treated in M BILE as kinetostatic transmission elements in the sense that they carry out signal processing tasks resembling the power transmission mechanism of the kinetostatic model Table 4 1 gives an overview of the currently supplied kinetostatic transmission elements Note that classes in MEJBILE can be either abstract or concrete For example the classes MoForceElement and MoChord are abstract while the classes MoConstantStepDriver MoChordPointPlane and MoLinearSpringDamper are concrete Abstract classes act merely as conceptual containers for the common properties of a familiy of elements They can not be instantiated directly and their implementation is thus of no interest for the user However one can introduce pointers to this type of objects that can be employed for accessing the generic properties of objects derived from this class On the other hand concrete classes represent actual implementations of entities that can be instantiated and used as many times as necessary This chapter introduces the some fundamental classes needed for kinetostatic modeling of tree type mechanical systems Other elements will be discussed in the subsequent chapters A detailed syntax description for these objects can be found in the appended MUIBILE 1 3 User s Guide 35 Section 4 2 Generic Properties of Kinetostatic Transmission Elements
119. leads to the end frame Kp of the lower segment e Branch III leads from base frame kK to end frame kK of the upper segment EL I Figure 5 9 Entities of interest for the topological types of measurement An overview of the currently supported topological types of measurement is given in Table 5 6 and Fig 5 10 The notation g Ky Ky in Table 5 6 denotes a scalar mea surement between any two frames Ky and Ky while the frame Kp represents the inertial frame The second column of Fig 5 10 represents the case in which the kinetostatics of 74 MUJBILE 1 3 User s Guide Section 5 2 Basic Properties of Measurement Objects branches I and II are passed to the measurement as a transmission chain This corresponds to the case of a self reconfiguring measurement as explained in Section 5 2 2 arguments measurement notes Kyy g Ko K absolute motion of reference frame K minus scalar y Ke KB Yy Kg Ke relative motion of Kp with respect to Kg minus scalar y Ky Ke g K z Ko difference of measurements of absolute motion of two g Kr Ko frames Kg and Kg Ko g g difference of two measurements one between the start Ke Kpg g and endpoint of a moving chain branch III and one based on the relative motion between two reference frames of the other segment the argument represents the transmission function of the moving chain see below Table 5 6 Types of mea
120. lied to a joint 52 scalar measurements 65 71 72 interlinking of 79 optimizing performance 77 Type I see scalar measurements ab solute motion type Type II see scalar measurements relative motion type Type III see scalar measurements dif ference of absolute motion type Type IV see scalar measurements dif ference of relative motion type scalar types 19 scalar variables accessing both types of 80 31 getting the type 30 self reconfiguring measurements 66 slider widget 107 solver objects transmission functions 60 solvers see constraint solvers source force applied and inertia 41 spatial measurements closure condition 70 rotational part 69 statics 71 spatial kinetostatic state objects 10 32 spatial load global and local 51 spatial measurements 65 basic kinematic formulas 70 basic static formulas 71 components of state 71 kinetostatics 69 types of 68 state objects 10 27 state of measurement 66 super loops 60 SYMKIN 58 target frame 72 topological types of measurement 65 transmission of forces 38 of motion 88 transmission subtasks 40 MUJBILE 1 3 User s Guide Section 8 4 MYIBILE for PC with Graphic User Interface mathematical terms 40 tree type systems 57 upper segment 74 USE_CORIOLIS 42 USE_INTERNAL 42 variable list 31 indexing 32 vector unit 73 vector components 23 38 vector products inner dyadic and vector 24 viewer 103 104 edit mode
121. loop structure see Fig 5 1 a tree type system b closed loop system Figure 5 1 Comparision of tree type and closed loop systems In systems featuring tree type structure there is one and only one path between any component and the inertial frame Thus the relative motions between any two pairs of neighboring bodies are independent and it is possible to process the kinetostatics of the elements on a component by component basis A user concerned with the modeling of such a system just needs to concatenate its components in an order that is compatible with its topological structure i e starting at the inertial system and ending at the tips of the branches When the bodies of the multibody system form closed loops the relative motions within the loop become dependent a change of relative motion at one place induces a change of relative motion at another place Such dependencies make it impossible to proceed joint by joint or body by body as in the tree type structure case Instead one has to formulate and solve so called constraint equations or closure conditions that hold the branches of the loop together In MUBILE the closure of loops is accomplished as a two stage process e In a first stage a set of characteristic measurements is defined whose van ishing indicates the closure of the loop These measurements also called chords in M ABILE are typically generalized distances between geometric elements such
122. lt lt 51 lt lt coupler_b mass elements SSSSSSSSSS S MoMassElement CouplerBody K6 m_d THETA_d d_a declare constraints tiii tiiit itii iiit MoChord3DPose core K1i0 left K7 right DO_ALL_BRANCHES where unknown DO_BRANCH_I where force declare implicit solver for constraint equations MoVariableList dependents dependents lt lt theta_2 lt lt theta_3 lt lt theta_4 lt lt CardanAngles MoImplicitConstraintSolver CoreSolver core dependents DependentChain declare a subsystem for the full solution of the loop SSS 2S2S2S222222 2222252222252 2522522522552 2525252252525 255252555 ES2SESS MoMapChain SolveAll SolveAll lt lt base base body MUIBILE 1 3 User s Guide 85 86 lt lt RightBranch move the input crank lt lt CoreSolver solve the implicit core lt lt CouplerBody compute D Alembert s forces at coupler list of independent coordinates MoVariableList q q lt lt theta_1 define parameter values beii ee ley 2 0 offset at the base r z 1 0 lenght of link 1 s z 2 0 lenght of link 2 d_a z 0 5 sqrt 5 0 half lenght of coupler d_b z 0 5 sqrt 5 0 half lenght of coupler m_d 1 0 mass of coupler THETA_d 1 0 symmetric inertia tensor for coupler generation of equations of motion and i
123. lt task From a topo logical viewpoint all joint variables are potential candidates for being selected as input variables However the possible occurence of turning points and limit e g stretched configurations limits the usefulness of some of these selections Luckily in most indus trial applications the places to be chosen as inputs are clearly marked by driving units or major motion directions However for some academic examples such a choice may be non evident or even impossible to make for a general simulation These cases require a careful assessment by the user This manual provides no instructions on how to choose appropriate input variables for a mechanism For this purpose the reader is referred to the supplied examples or for more difficult cases to the specialized literature 5 1 1 Example Inverse Dynamics of a Spatial Shaker Mechanism For illustration of the just mentioned concepts a MJJBILE model for the inverse dynam ics of a simple multibody loop is regarded The objective is to compute the motion of the system at a prescribed configuration of the input variable s and the input torque s 60 MUJBILE 1 3 User s Guide Section 5 1 Basic Methods of Formulating Loop Closure Conditions required to drive the mechanism at this position at a given constant speed The corre sponding MUIBILE code is reproduced below and shall be commented in the following paragraphs The model makes use of some classes that are new to t
124. motion of the system to the screen There are also other routines for integration installed in MIJBILE Examples hereof are the explicit Euler method MoExplicitEulerIntegrator or the Runge Kutta method MoRungeKuttaIntegrator In M BILE numerical integration as well as other com putationally expensive numerical tasks are solved using modules of standard numerical libraries Currently there exist interfaces for the numerical libraries SLATEC NAG and IMSL However only the SLATEC routines are actually shipped with the MUIJBILE soft ware The libraries NAG and IMSL are liable to licenses which have to be purchased by the user directly from the corresponding dealers The user can freely choose between these methods and attach them to the mechanical models without having to regard the details of numerical algorithms 2 3 Summary The example discussed above displays some of the capabilities of the object oriented multibody modeling library MIXBILE It can be appreciated that objects of MIJBILE provide an intuitive and natural language for rapid prototyping of mechanical systems that is also well suited for devising hand tailored programs for simulating systems Hand tailored programs have the advantage of being open and easy to extend Thus once created models can be extended as the demands grow This is the key for efficient and integrated approaches featuring code reusal and interdisciplinary procedures as pursued by this package 18
125. mper joint based spring damper element MoFrame KJointi KJoint2 MoAngularVariable theta MoElementaryJoint R KJoint1 KJoint2 theta MoLinearSpringDamper spring2 R k c 4 4 2 Mass Elements Mass elements model the inertia properties of a rigid body i e its mass m and its inertia tensor Oc In MUBILE the inertia tensor is assumed to be defined with respect to the center of gravity C of the body The center of gravity itself can be offset from the origin of a reference frame K by a vector As as depicted in Fig 4 8 As with the components of MoFrame all tensorial quantities are always assumed to be decomposed with respect to the actual frame K As m Os T e a mechanical model b object oriented model Figure 4 8 Model of a mass element Theoretical background Modeling of inertia forces In ME3IBILE 1 3 inertia properties are modeled as d Alembert s forces Under a general motion of the frame K the d Alembert s forces exerted by the mass upon the origin of the frame K are f mla wx As w x w x As T Oswt wxOgw Asxf Note that in contrast to the force element the force of the mass element depends also on the accel eration of the frame to which it is attached However this dependency is only linear A general mass element takes as arguments e the frame to which the mass properties are to be attached MUIBILE 1 3 User s Guide 53 Section 4 5 Modeling of a SC
126. nclude lt Mobile MoVector h gt main MoVector v u 0 0 1 w MoVector 1 1 0 veFutw MoVectora 3 Vv MoVector b vA vhw a w u cout lt lt b LWEN prints to standard output 4 2 3 lt lt a y lt lt prints to standard output 3 lt lt u w lt lt n prints to standard output 0 lt Newline gt Note the different ways in which a vector can be defined Note also the way operator precedence can be employed in order to reduce the number of parenthesis 3 3 2 Matrices Three dimensional matrices come in different flavors in MIJBILE there are matrices representing rotations and matrices respresenting inertia properties Similarly to the definition of angular and linear scalars these two categories of objects although they look alike differ considerable in their inner structure while rotational matrices are orthogonal inertia matrices are always positive definite Thus in MI BILE special types of matrices are introduced that take account of matrix structure and allowed operations for each type Fig 3 2 displays the functional hierarchy for the matrices defined in the MIJBILE package In this hierarchy types to the right support all operations that the types to the left do This hierarchy is only conceptual It is not actually implemented in this manner In particular the data of the three elementary rotation matrices is structured in a different 24 MUJBILE 1 3 User s Gu
127. nd Symbol Conventions Throughout this manual the following syntactical and lexicographic conventions are used e Program listings examples and outputs are rendered in small courier font For example a program fragment describing a simple pendulum and calculating and printing the position of its center of mass is include lt Mobile MoMapChain h gt include lt Mobile MoRigidLink h gt include lt Mobile MoElementaryJoint h gt main MoFrame Ki K2 K3 MoVector v 1 0 1 MoAngularVariable beta MoElementaryJoint R1i K1 K2 beta MoRigidLink L1 K2 K3 v MoMapChain Pendulum Pendulum lt lt R1 lt lt Li beta q PI 2 Pendulum doMotion cout lt lt Position lt lt K3 R K3 r lt lt n MUIBILE 1 3 User s Guide 5 Section 1 7 Style and Symbol Conventions After compiling the program one can execute the code and obtain the result like this Pendulum Position 0 0 1 0 1 0 _ e Class names and keywords in syntax descriptions are typed in courier font as for example in MoVector name Class names and keywords must be typed exactly as shown e Identifiers i e variable names are printed in slanted courier font You can replace the names by any character string allowed as an identifier e Types of arguments passed to functions are typeset in lt italic courier gt enclosed by angle brackets Replace these entries by a permissible identifier or variable value of the
128. nding on the objectives of the simulation For example users seeking a high degree of efficiency need to access closed form solutions where possible in order to avoid redundant computations while users requiring a rapid yet maybe not so efficient modeling are satisfied with it erative solution procedures At present M JBILE provides only the basic constituents for implementing the different loop closure and solution strategies Users must select the ones that most closely fit their needs and concatenate them to appropriate transmis sion chains Automatic loop closure and resolution strategies will be installed in future versions of MEJBILE At present the user is referred to the dedicated CA Computer Algebra program SYMKIN which is a symbolical manipulation program written in Mathematica that automatically eliminates redundant computations and produces closed form solutions where possible This chapter describes the basic mechanisms for generating and solving constraint equa tions with MIJBILE Some sections particularly those concerning scalar measurements and closed form solutions are quite involved and may thus cause difficulties of understand ing to the casual reader However these sections describe advanced solution techniques that can be skipped when only basic modeling features are required Casual readers are adviced to only browse through these sections in order to grasp the basic ideas and to apply the universal better und
129. ne DO_BRANCH_I where unknown DO_BRANCH_II DO_BRANCH_III where force DO_BRANCH_III input MoChordPointPlane compl_2b compl_2a all references as in zAxis compl_2a explicit solvers for constraint equations SSRSSSSSSSSSssSsssssssssssssssssssssssss MoExplicitConstraintSolver CoreSolver core theta_4 MoExplicitConstraintSolver FirstHooke compl_1 theta_2 MoExplicitConstraintSolver SecondHooke compl_2a compl_2b theta_3 FirstHooke selectBranch mass element MoReal m_d MoInertiaTensor THETA_d MoMassElement CouplerBody K7 m_d THETA_d d_s create a subsystem for applying the input motion kiii MoMapChain input input lt lt 1_1k lt lt R1 lt lt r_lk create a map chain for the complete kinematics of the loop sssneesssssanssssssssnssesssanssesssasssssssassssssassssss MoMapChain SolveAll SolveAll lt lt input move the input crank lt lt CoreSolver solve the implicit core lt lt FirstHooke solve the first hooke angle lt lt SecondHooke solve the second hooke angle lt lt CouplerBody compute D Alembert s forces at coupler list of independent coordinates 92 MUJBILE 1 3 User s Guide MoVariableList q q lt lt theta_i generation of equations of motion and integration SSS 2525252522222 2225252525255 2525225252252 25252525252 ES
130. nent position orientation frame R MoRotationMatrix translation frame r MoVector velocity angular frame ang_v MoVector linear frame lin_v MoVector acceleration angular frame ang_a MoVector linear frame lin_a MoVector load moment frame t MoVector force frame f MoVector Table 3 12 Subentries for kinetostatic state objects of type MoFrame For example if K is an object of type MoFrame the corresponding location of the origin with respect to the fixed frame can be printed as cout lt lt K R K r Note that here the origin vector is premultiplied by the rotation matrix of the frame prior to being printed Thus the printed components are measured with respect to the fixed MUIBILE 1 3 User s Guide 33 Section 3 4 State Objects frame Conversely if one has a vector decomposed with respect to the fixed frame and wants to apply it the actual frame one transforms it to the actual frame by K lin_v vector K R Note that here the vector is post multiplied by the rotation matrix This is equivalent to pre multiplying the vector with the transpose of the matrix K R see Section 3 3 As it was the case with scalar state objects it is possible to create logical units comprising several frames either by arrays or by lists In M IBILE lists of frames are created as instances of the class MoFrameList Objects of this class are used in exactly the same manner as objects of type MoVariableList see Section 3 4 1 Which of th
131. nerates nor consumes power Then equality of virtual work at the input and output holds and it follows 38 MUIBILE 1 3 User s Guide Section 4 2 Generic Properties of Kinetostatic Transmission Elements After substituting oq Js q and noting that this condition must hold for all q R the force transmission function Q H A follows If the kinetostatic transmission element is not ideal it contributes to the ideal transmission of forces an additional term Q which represents the source force of the element Such source forces can result for example from friction effects but can also model contributions due to applied forces or mass effects With the source force Q the transmission of forces takes on the form force Q J Q Q 4 2 Note that in general the Jacobian J is not quadratic so for most transmission elements this relationship cannot be reversed Thus the natural direction of transmission of forces is in opposite direction to the motion mapping as was already depicted in Fig 4 2 4 2 2 Invoking Motion and Force Transmission The transmission of motion and force is implemented for each type of mechanical compo nent in a particular and efficient manner The user can invoke these functions either by appending the strings doMotion and doForce to the name of the kinetostatic trans mission element or by using the pointer dereferencing mechanism pointer gt doMotion and pointer gt doForce Moreover
132. nly informative and need not be regarded by the casual user MUIBILE 1 3 User s Guide 43 Section 4 3 Joints Links and Chains 4 3 1 The Object MoRigidLink A rigid link can be thought of as a transmission element that transports a coordinate frame K via a constant rotation AR and a constant displacement Ar to another frame K Fig 4 4 Here AR represents the orthogonal matrix transforming coordinates with respect to K to coordinates with respect to K and Ar is the vector from the origin of K to the origin of K decomposed in K 1 AR g La K K Za r R r R b object oriented model Ko a mechanical model Figure 4 4 Model of a rigid link Rigid links are modeled in MUJBILE as instances of class MoRigidLink The typical definition of the rigid link consists in stating the input frame the output frame and the relative displacement and rotation matrices as arguments to the rigid link object An example is MoFrame K_Input K_Output MoVector Delta_r MoRotationMatrix Delta_R MoRigidLink Link K_Input K_Output Delta_R Delta_r Here Link is the name of the newly introduced object representing the rigid link and K_Input and K_Output are the input and output frames respectively The values of the relative displacement Delta_r and relative rotation Delta_R need not be specified at this point The object rigid link remembers their location in storage spa
133. nputs of the loop 3 create one or more object s modeling the dependent chains of the dissected loop each dependent chain is typically an object of type MoMapChain containing the kinetostatics from the dependent variables to the cut frames 4 create one or more object s derived from type MoChord that describe the loop closure condition s 5 create an object of type MoSolver passing to it the dependent chain s the list of dependent variable s and the object representing the closure condition s After carrying out these steps the user can employ the resulting object of type MoSolver as a simple kinetostatic transmission element representing the kinetostatics of the closed loop s The doMotion function of the solver generates the motion of the dependent chains so that they follow the input motion while keeping the loop closed the doForce function computes the forces at the cut frames and within the dependent chains so that static equilibrium is achieved Usage of this object is then fully equivalent to the usage of any other kinetostatic transmission element such as an elementary joint or a rigid link i e solver objects can be used again as constituents of chains of kinetostatic transmission elements or even super loops exhibiting in their branches other loops Note that the choice of independent and by this also of dependent variables for a given loop is in general a non uniquely solvable and in some cases difficu
134. ns defined for vectors By making use of these operators not only one can write programs resembling common vectorial expressions but it is also guaranteed that the operations are carried out in the most efficient way The components of each vector are represented by the member elements x y z of type MoReal One accesses these components by appending x y or z to the name of the vector MUIBILE 1 3 User s Guide 23 Section 3 3 Vectors and Matrices Operator usage Action vector vector vector vector vector vector vector vector vector scalar vector vector vector vector vector vector assign a vector vector change sign of vector vector add two vectors vector subtract two vectors real create inner product of two vectors vector multiply vector by scalar vector create vector product left times right vector add and assign a vector vector vector vector subtract and assign a vector vector h vector vector vector product and assign left times right vector XYZrotation gt vector transform by elementary rotation and assign vector X YZrotation gt vector transform by transpose of elementary rotation and assign vector MoNullState gt vector assign zero vector a ea bead fonds oh Table 3 5 Overview of operations for objects of type MoVector The following program fragment shows some examples of the use of vectors include lt iostream h gt i
135. nt equations i e for the treatment of closed loops the generation of direct dynamics and for the numerical solution of the equations of motion will be discussed in subsequent chapters 4 1 Overview of Supplied Kinetostatic Transmission Elements The mechanical building blocks in M 4BILE are particular specializations of a generic element denominated MoMap This super ancestor subsumes the generic properties of mechanical components in form of virtual functions that are guaranteed to be supplied by all mechanical components whatever the details of their implementation Two of these functions are doMotion for the transmission of motion and doForce for the transmission of forces Modules of mechanical components are obtained in MEYBILE by derivation from the class MoMap The idea behind this kind of module organization is that one can address the generic properties of a component without knowing the exact type of implementation hidden behind it In MI3BILE one can access the generic properties of any mechanical component e g a joint a rigid link or a complete vehicle suspension by regarding them as instances of class MoMap Fig 4 1 illustrates the class hierarchy for a part of the family of kinetostatic transmission elements Note that some nodes act again as a base class for several descendants Note also that some elements like MoOOutputGenerator MoIntegrator and MoDriver are not actually mechanica
136. ntPointLinear chord_IIIb KEprime KE phi Type IV MoChordPointPointLinear chord_IVa KEprime phiPrime KE KB MoChordPointPointLinear chord_IVb KEprime phiPrime KE KB phi Note that it is not necessary to pass the base frame of the upper segment KB to the chord This frame is extracted internally from the transmission element coupler during initialization of the object chord 5 2 7 Optimizing Performance by Specification of Active Branches Measurement objects provide some rudimentary mechanisms for optimizing performance in the calculation of their outputs The basic idea is to tell the chord at which of the MUIBILE 1 3 User s Guide 77 Section 5 2 Basic Properties of Measurement Objects possibly three existing branches I II III it is necessary to apply an action as e g motion transmission or force application For example if the base frame Kg is kept fixed though not necessary congruent to the inertial frame the measurement object does not need to upgrade that information and it can reuse the values computed at initialization time Optimization information does not have any effect on the accuracy of the results but can lead to up to 20 of improvement in execution time during solution of the constraint equations Users not concerned with computational performance issues can skip this section Optimization information is passed to the measurement object by up three optional ar guments app
137. ntegration gt MoMechanicalSystem SystemDynamic q SolveAll KO zAxis MoAdamsIntegrator SystemIntegrator SystemDynamic MoReal dt 0 01 SystemIntegrator setTimeInterval dt initial conditions kie theta_1 q 1 0 DEG_TO_RAD theta_2 q DEG_TO_RAD 0 0 theta_3 q DEG_TO_RAD 0 0 theta_4 q DEG_TO_RAD 40 0 theta_5 q DEG_TO_RAD 20 0 theta_6 q DEG_TO_RAD 0 0 theta_7 q DEG_TO_RAD 0 0 for int i 0 i lt 100 SystemIntegrator doMotion MUJBILE 1 3 User s Guide R4 theta_4 R1 theta_1 R2 theta_2 R3 theta_3 a closed system b opened system CouplerBody K7 K8 d_lk sphere FA K3 rk A PK Lk G ki K2 theta_1 theta_3 theta_2 c iconic model Figure 5 13 Modeling of the Dynamics of a Shaker Mechanism 5 4 2 Joint Assembly of a Shaker Mechanism include lt Mobile MoElementaryJoint h gt include lt Mobile MoRigidLink h gt include lt Mobile MoImplicitConstraintSolver h gt include lt Mobile MoMassElement h gt include lt Mobile MoAdamsIntegrator h gt main reference frames MUIBILE 1 3 User s Guide 87 state variables 3 5 5 e SSS Sd MoAngularVariable theta_1 theta_2 theta_3 theta_4 joints MoElementaryJoint R1 Ki K2 theta_1 yAxis MoElementaryJoint R2 KO K4 theta
138. oDynamics h all of Table 1 4 Inventor MoGraphics h all of Table 1 5 Table 1 6 Container header files MUIBILE 1 3 User s Guide 9 2 Overview This chapter is devised as an introduction to the capabilities of MEJBILE The reader will be guided through the process of modeling the dynamics of a simple example starting from the basic topological structure and ending with the generation of an animation The intention is to display the fundamental ideas underlying the MIJBILE philosophy together with a desription of the basic modeling steps It is thus not necessary to understand all the underlying mechanisms at this point The details are discussed in the subsequent chapters 2 1 Structure of MIJBILE One of the main features of MIIBILE is that it allows the user to model mechanical sys tems as executable programs that can be used as building blocks for other environments This is achieved by representing each real world component by a dedicated object that is capable of performing some well defined set of actions upon request The objects of MYBILE are roughly organized in three categories a basic mathematical objects which provide the algebraic resources for performing the typical multibody calculations b kinetostatic state objects which are used to store and retrieve kinematic or load related information at specific locations of the multibody system c kinetostatic transmission elements which transmit the information stored wi
139. obal wrenches are useful for modeling effects such as gravitational forces although the objects MUIBILE 1 3 User s Guide 51 Section 4 4 Force and Mass Elements for generation of dynamical equations provide an alternative and more efficient way of accomplishing this Local wrenches are defined with respect to the actual frame They allow the user to model forces that follow the moving frame The following code fragment illustrates the creation of constant spatial loads with respect to the inertial and actual frames respectively MoFrame Ki K2 MoVector f n MoConstantWrench moving K1 f n LOCAL moving force amp torque MoConstantWrench fixed K2 f n GLOBAL fixed force amp torque The objects moving and fixed are instances of an object applying a spatial load The spatial load consists in both cases of a force f and a moment n In the first case the components of f and n are interpreted to be given with respect to K1 In the second example the elements of f and n are interpreted to be given with respect to the inertial frame If no fourth parameter is specified it is given the default value LOCAL Scalar loads are generated by instances of class MoLinearSpringDamper Objects of this type allow the user to apply a load which is a linear function of the displacement and the velocity of a scalar variable i e which models a classical spring damper element The element c
140. objects always takes an additional argument of type MoMap embodying this isolated chain Objects of this type are useful for establishing triangular systems of constraint equations which are recursively solvable Such cases can be found in abundance in technical systems but the corresponding methodology is very complex and not suited for casual users Thus these objects are not intended for users who are seeking rapid prototyping solutions to their problems Such users may skip the following sections and go right to Section 5 3 where the solution of constraint equations is described The use of scalar measurements of type IV for efficient solution of constraint equations will be further discussed in Section 5 3 2 The following code fragment illustrates the initialization of measurement objects for the four topological types described above As an example the linear measurement between two points is employed However any other scalar measurement could have been employed instead Note that spatial measurements support only constructors with two reference frames MoFrame KE KB KEprime MoMapChain phi phiPrime MoLinearVariable y Type I MoChordPointPointLinear chord_Ia KE y MoChordPointPointLinear chord_Ib KE y phi Type II MoChordPointPointLinear chord_IIa KE KB y MoChordPointPointLinear chord_IIb KE KB y phi Type III MoChordPointPointLinear chord_IIIa KEprime KE MoChordPoi
141. of the system and attaches to it subsequently the mass elements at desired places 2 2 2 Calculating Dynamic Properties The code discussed above represents only a basic skeleton describing the kinematics and statics of the system Based on this model further computations can be performed One example is the generation of the equation of motion which is discussed next Theoretical background Equation of motion for a one degree of freedom system The system at hand has only one degree of freedom and no damping effects occur Thus the dynamics of the system are governed by the scalar equation of motion m 6 b Q Here m is the generalized mass b is the generalized Coriolis and centrifugal force and Q is the generalized applied force of the system Equations of motion are generated in M BILE by objects of type MoEqmBuilder These builder objects take a mechanical model represented by a kinetostatic transmission element and a set of variables acting as generalized coordinates and compute the cor responding mass matrix and vectors of generalized Coriolis and applied forces from this information Further arguments can be passed to the builder of equations of motion that describe the reference frame acting as the inertial frame and the upwards direction i e the direction opposite to gravitation For the pendulum example derived above the corresponding code has the following ap pearing 14 MUJBI
142. olving relative velocities and not relative accelerations The subtasks related to the forces are collected in Table 4 5 Note that here two types of source forces are discerned namely i applied source forces and ii inertia source forces Applied source forces are those generated by springs frictional effects etc Inertia source forces are those generated by mass elements Setting the subtask selection parameter to DO_INTERNAL computes both of these forces The distinction between the two types of forces stems from the M JBILE algorithm for the generation of dynamical equations which involves calculating separately the mass matrix and the vector of applied forces This is discussed in more detail in Chapter 6 Similarly to the quadratic velocity terms in the acceleration the internal forces can be computed only COMPUTE_INTERNAL or transmitted only TRANSMIT_INTERNAL or both computed and transmitted DO_INTERNAL This is done to optimize computations when the evaluation of force terms of the motion state is very time consuming In most cases the selection of the appropriate subtasks is performed by the transmission elements automatically so the user does not need to be concerned about choosing the correct values for the transmission subtask selection parameter However direct use of the transmission subtask selection parameter gives to the user better control over the scope of the computations and by this a
143. on invokations can be set via the member function setAnimationIncrement time of the viewer where time is a real number representing the time interval If the computation of the doMotion functions takes less time than the given time inter val the system will wait until the required interval elapses If the computation of the doMotion functions takes more time than the given time interval the system will repeat the doMotion call immediately progressing at maximum computational speed The user can force the viewer to work with highest possible speed by specifying a zero value for the animation time interval Animation is started from the viewer by selecting Run Animation from the drop down menu under the entry Anim at the right of the main menu bar Animation is stopped by selecting Run Animation from the drop down menu under the entry Anim at the right of the main menu bar Animation can be used in conjunction with manipulators and or 110 MUJBILE 1 3 User s Guide Section 7 5 Further Animation Capabilities sliders Note that attempting to start an animation without supplying appropriate ob jects through the addAnimationObject member function will result in unpredictable errors When animating a numerical integration process one can stop the animation and re configure the mechanism parts by operating the manipulators and sliders Selecting Run Animation again will th
144. onditions initCond addSlider phi q 360 360 angle initCond addSlider phi qd 10 10 angular vel Scene show parameters show initCond show return MoScene mainLoop The program code has only two differences in comparision to the code in Page 116 The first one is that the class MoScene has no member function makeManipulator joint The second one is that the function call Scene makeShape K2 MoSphere so has been replaced by Scene makeShape K2 SPHERE 0 1 MUIBILE 1 3 User s Guide 119 Section 8 4 MUIBILE for PC with Graphic User Interface where the value 0 1 indicates the radius of the sphere For compilation select now mobile home 1lib 1ibOpenGL in Step 4e of the general procedure for generation and execution of the model displayed in Page 116 see Fig 8 3 Compilation of the program can then be performed in the same manner as in Section 8 2 For starting the MIIJBILE program execute the file MobileGL exe in the directory mobile home mobile EXE Again a window appears where the MJJBILE model can be selected see Fig 8 5 mobile Input Dialog RES Suchen in ja Debug 7 El c it Dateiname Pendulum dl DLL Start Dateityp MobileGL linkfile dll Interactive choose a Mobile Model dll Figure 8 5 Start of a model with OpenGL 8 4 MUIBILE for PC with Graphic User Interface When starting MobileGL exe from the directory mobile
145. ow see Fig 8 4 select the directory and MUJBILE model name with the d11 extension in our example Pendulum d11 Operate the mechanical model in the same manner as on UNIX environments 116 MUJBILE 1 3 User s Guide Section 8 3 MIIBILE for PC with OpenGL 8 3 MOBILE for PC with OpenGL Graphic Interface The OpenGL Graphic Interface frees the user from the burden of purchasing an Open Inventor license However please note that in the OpenGL version not all sophisticated features of Open Inventor will be available For example it is only possible to render M BILE default representations of objects and not CAD files New projects are generated with the OpenGL graphics environment in the almost in same way as with the Open Inventor version The only difference is that in the Projects window Mobile OpenGL AppWizard has to be selected instead of Mobile Inventor AppWizard as explained in Page 114 Fig 8 2 Please read the instructions in Section 8 2 before continuing as this section specifies only the differences to the Open Inventor procedure The generated file Pendulum cpp for the OpenGL version is include Mobile MobileGL MoMFC h include Mobile MobileGL MoScene h include Mobile MobileGL MoWidget h _void main insert here your Mobile code return MoScene mainLoop Again do not change any of these lines as otherwise the model will not compile and function correctly You may type in again a MU
146. r the different types of spatial measurements currently supplied with M3BILE MoChord3D element Position Pose state 0 3 Ar x state 1 Ar y state 2 Ar z state 3 state 4 state 5 state 0 state 1i state 2 state 3 state 4 state 5 NQHNS amp state 0 state 1i state 2 state 3 state 4 state 5 NEHRING state 0 state 1 Q state 2 state 3 state 4 state 5 NGHR NGS Table 5 4 Elements of the array state for spatial measurements 5 2 5 Scalar Measurements Scalar measurements generate projections from spatial frames to real numbers The basic idea of this projection is illustrated in Fig 5 7 in its most simple form The measurement MUIBILE 1 3 User s Guide 71 Section 5 2 Basic Properties of Measurement Objects object takes the motion of two frames termed the target frame Kg and the base frame Kpg and produces a scalar quantity that depends only on the relative motion between both frames An example of such a measurement is the distance between the origins of the frames depicted in Fig 5 7 Figure 5 7 Basic form of a scalar measurement Scalar measurements can exhibit almost any combination of geometrical topological and activity type currently supplied with the MSXBILE software Only the segment assembly topological measurement type always requires that the measurement is self reconfiguring M IBILE currently supports five types of scala
147. r geometric measurements all of which arise from the combinations of the two geometric elements point and plane T2 The quadratic distance between the origins of two frames This class is suited for formulating constraints between two spherical joints or a Hooke joint and a spherical joint class name MoChordPointPointQuadratic The linear distance between the origins of two frames This class is suited for generating linear springs but not so well suited for constraint formulation due to its poor computational performance class name MoChordPointPointLinear the cosine of the angle between two coordinate planes This class can be employed for resolution of variables in a spherical joint class name MoChordPlaneP lane The distance from a point to a plane where the point is located at the origin of frame Ky and the plane is coplanar to a coordinate plane of frame Kg This measurement is suited for resolving angles at Hooke joints when the second joint is a spherical joint one chooses as normal vector the unit vector in direction of the second axis of the Hooke joint and as point the center of the spherical joint eliminating by this the second joint variable of the Hook joint and the three joint variables of the spherical joint class name MoChordPointPlane The shortest distance from a plane to a point where the plane is now a coordinate plane of frame Ky and the point is the origin of frame Kg
148. rames and map them to the corresponding scalar or vectorial quantities of the measurement In order to reduce the modeling effort MfJBILE also allows the user to include a transmission chain in the measurement that is invoked each time the measurement object is traversed Such measurements are termed self reconfiguring measurements For an illustration of this concept consider the task of measuring the vector from a moving camera to the end effector of a robot Fig 5 5 Let the motion of the robot be carried 66 MUJBILE 1 3 User s Guide Section 5 2 Basic Properties of Measurement Objects endEffectorFrame camera trajectory manipulatorChain A approachVector 1 cameraFrame inertial frame cameraChain Figure 5 5 Example of a measurement for a moving object out by a transmission chain manipulatorChain while the motion of the camera is produced by the object cameraChain In order to measure the actual motion the user first has to invoke the doMotion functions for the manipulator chain and the camera chain respectively and then that of the measurement object A corresponding program fragment might look like this MoFrame endEffectorFrame MoFrame cameraFrame MoMapChain manipulatorChain robot kinematics MoMapChain cameraChain camera motion MoVector approachVector difference vector definition of measurement object MoChord3DPosition simpleChord endEffectorFrame cameraFr
149. ranslation and rotation with a constant pitch i e coupling between these two and MoCylindricalJoint for a combination of independent translation and rotation with respect to the same axis As a transmission element the elementary joint maps the motion of an input reference frame K and the value of the joint variable 8 to the motion of an output frame K Fig 4 5 Depending on whether the joint variable is an angle 8 O or a displacement 8 s the joint becomes revolute or prismatic For the case of a cylindric or screw joint both types of variables are active The joint axis is assumed to correspond to one of the coordinate axes of K For the definition of a joint in MIIBILE one specifies the reference frames at the input and output the joint variable and optionally an argument indicating which coordinate axis to use for the transformation The type of joint i e whether it is prismatic or revolute is recognized by the type of the variable passed as an argument For example the following code fragment initializes a prismatic joint 46 MUJBILE 1 3 User s Guide Section 4 3 Joints Links and Chains 8 D 7 i E a mechanical model b object oriented model Figure 4 5 Model of an elementary joint MoFrame K_Input K_Output MoLinearVariable s MoElementaryJoint Slider K_Input K_Output s prismatic joint while the next one produces a revolute joint MoFrame K_Input K_Output MoAngularVariable Th
150. rator MoAdamsIntegrator and MoRungeKuttalIntegrator Below the main properties of these classes are described 6 1 The class MoEqmBuilder Objects of class for MoEqmBuilder are responsible in M YBILE for generating the dy namical equations of mechanical systems These dynamical equations are determined by computing then inverse dynamics of the system repeatedly with changing input values for velocities and accelerations A short description of the underlying mathematics is included below for easier reference Theoretical background Generation of the Equations of Motion The transmission elements described in the previous chapters give a simple means of generating and transmitting motions and loads within a multibody system Consider a mechanical system having f independent generalized coordinates q q q 1 and let the transmission of these coordinates to the mass and the force elements be given by a transmission element yg denoted global kinematics Let also the global kinematics be decomposed in a first part denominated here the kinematical subsystem which encloses the chains of links and joints of the system including closed loops and a second part where the set of mass and force elements are subsequentially defined as leaf elements Fig 6 1 In this way the overall system is partinioned in a skeleton containing the pure kinematostatic transmission structure of the system and additional elements producing force
151. rement measurements generating tenso rial quantities are hereby denoted by spatial measurements while measurements producing scalar outputs are termed scalar measurements B the topological type which is determined by the number of frames involved in the measurement as well as the type of motion relative or absolute regarded in the measurement and C the activity type which characterizes the behaviour static or self reconfiguring of the measurement with respect to the motion of the involved frames MUIBILE 1 3 User s Guide 65 Section 5 2 Basic Properties of Measurement Objects Measurement objects in MZYBILE can combine almost any pattern of attributes from the three basic types listed above This results in a quite large number of potential measurement objects Table 5 1 lists the geometric types of measurements currently supplied with the M ABILE software The measurements described there are described based on the topological type involving only two frames For scalar measurements other topological types exist which involve up to four frames This is described in more detail in Section 5 2 5 The two activity types of measurement are explained in Section 5 2 2 type of measurement action when applied to two reference frames MoChord base class for chord elements MoChordPlanePlane cosine of the angle between two coordinate planes MoChordPlanePoint shortest distance from a base plane and a target point MoChordPointPointQuadrati
152. rmal kinetostatic transmission elements The only difference to the previously introduced components is that force and mass elements have no kinematical output i e that they act as leaves attached to the rest of the system One can imagine the multibody system as consisting on the one side of a kinematic skeleton in which the interconnection structure is defined by massless links joints etc and on the other hand of sources of dynamic effects like forces and inertia which are added on to the kinetostatic skeleton 4 4 1 Force Elements Force elements can produce either scalar or spatial loads Currently only two force elements are supported with the MEABILE software One of these force elements is suitable for applying a spatial load whose components are measured either with respect to the inertial frame or with respect to the actual frame The other generates scalar forces either within joints or between frames Spatial loads are generated by objects of type MoConstantWrench These objects allow the user to apply a wrench i e a force and a moment at an arbitrary reference frame of the system The moment is always interpreted to be acting at the origin of the frame However the components of the force and moment can be interpreted in one of two ways One can define global wrenches which are defined with respect to the inertial frame and local wrenches which are decomposed with respect to the actual reference frame Gl
153. rques that are necessary to achieve a prescribed motion of the robot links i e the so called inverse dynamics of the system The system is depicted in Fig 4 9 It consists of three parallel revolute axis and three rigid links The axes of the revolute joints are directed upwards in positive z direction The frame KO represents the inertial reference frame All links have equal masses and 54 MUJBILE 1 3 User s Guide Section 4 5 Modeling of a SCARA robot R3 beta_3 R1 beta_1 R2 beta_2 KO a mechanical model K3 K4 link_2 R2 beta_2 OX fx G R3 VN beta_3 K1 link_1 M2 Be beta_1 ANAN K2 K5 ri G A Figure 4 9 Modeling of the inverse dynamics of a SCARA robot M1 M3 K b iconic model moments of inertia The last link is modeled only by its mass properties as its geometric dimensions are immaterial for the present analysis Below the corresponding MEJBILE code is reproduced The inertia tensor is defined by its diagonal entries Moreover the center of gravity of the links are offset by vectors p1 p2 and p3 from the origins of the frames of attachment Note that the inverse dynamics model is just obtained by appending the mass elements to the kinetostatic chain representing the kinematic skeleton of the system include lt Mobile MoElementaryJoint h gt include lt Mobile MoRigidLink h gt include lt Mobile MoMassElement h gt include lt Mobile MoMapChain h gt void main
154. s is of no importance It is only important to put them in correct MUIBILE 1 3 User s Guide 13 Section 2 2 Example Analysis of a Simple Pendulum sequence into the composite chain Moreover the values of the components vectors and variables are not defined at this point Only the topological structure is memorized during the definition of the objects This is due to the fact that variables are passed by reference in the constructors of MIJBILE Thus only addresses of the arguments are stored in contrast to the pass by value technique in which the actual value of the variables is employed In MIYBILE values are re read each time a motion or force traversal of the system is carried out during simulation This gives a certain degree of symbolic capabilities to models established with MJIBILE The mass property is defined as an additional transmission element named Tip At first sight this seems redundant why aren t mass properties defined directly for the link The background is that many mechanical systems can be modeled as massless skeletons for which mass endowed parts occur only at discrete locations In this case a lot of redundant calculations would be carried out if these masses are set numerically to zero For this reason properties of motion and force transmission have been separated in MUQBILE from inertia features The user first models a massless scaffolding representing the overall interconection structure
155. s of these two planes is equal to the unit vector in direction of the axis of action of the angular variable i e if the unknown rotation is about the y axis the first measurement should be with respect to the plane normal to the z axis and the second with respect to the plane normal to the x axis An example of this type of solution displayed in Section 5 4 3 80 MUJBILE 1 3 User s Guide Section 5 3 Objects for Solving Constraints 5 3 Objects for Solving Constraints Constraint solving objects in MUIBILE can process the kinetostatics of one or more closed loops In order to keep the loops closed a solver object needs three pieces of information i the measurements whose vanishing will signal the closure of the loops ii the depen dent variables whose variation will lead to the closure of the loop and iii the dependent chain that will reconfigure the cut frames involved in the measurements after perturbing the dependent variables Currently there are two types of solver objects installed A explicit solvers which can resolve a scalar constraint equation explicitly in terms of one unkown class name MoExplicitConstraintSolver and A implicit solvers which can resolve any number of constraint equation iteratively for a set of unknowns class name MoImplicitConstraintSolver As was explained in Section 5 1 there exist three basic methods for establishing constraint equations in closed loops C1 cutting the loop at on
156. s the notion of the kinetostatic transmission element an abstraction which describes the minimal basic services that any mechanical component must provide Consider a mechanical system consisting of several hundred of pieces ranging from bolts over joints to complete subsystems such as gear boxes transmission elements suspension mechanisms etc In real world the designer will typically strive to use standardized simple interfaces that make possible a rapid assembly and disassembly of the pieces This makes the construction easily maintainable and extendable In M MBILE this idea is carried over to the modeling of multibody systems by employing as interfaces for the models not only the data but also the functions By this measure the user does not need to peek into the core of the models to obtain information about internal implementation details but can construct a global model by making only use of abstract services Hence a model can be completed incrementally by adding or removing parts without affecting MUIBILE 1 3 User s Guide 37 Section 4 2 Generic Properties of Kinetostatic Transmission Elements the rest of the operational model The transmission behaviour or the services of a kinetostatic transmission element can be best described by considering a simple element mapping a set of n scalar kinematical input variables q R to a set of m scalar kinematical output variables q R as displayed in
157. s to Inventor format via the function call mobile home dir bin DxfToIv AutoCad file dxf Inventor file iv Translating from AutoCad dxf format to Inventor format allows the user to employ real istic geometric information for the animation 7 3 Prescribing Motion by Sliders Apart from manipulators MZJBILE provides also the capability of prescribing user feed back through user defined sliders User defined sliders are grouped in objects of type MoWidget termed slider widgets Slider widgets are placed in own windows that can be located anywhere on the screen A slider widget is initialized with the name of the viewer to which it is to be attached a kinetostatic transmission element and a character string The kinetostatic transmission element is traversed in motion mode each time a slider of the widget is actuated The character string is displayed as the title of the slider widget One can define several slider widgets for the same viewer For each slider widget one can define severl sliders by invoking the functions MUIBILE 1 3 User s Guide 107 Section 7 3 Prescribing Motion by Sliders slider widget addSlider real title min max slider widget addSlider angle title min max Here real and angle are names of objects of type MoReal and MoAngle used as input variables or parameters for previously defined kinetostatic transmission elements The numbers min and max
158. sess the implementation files The header files are shipped with every license of MIJBILE The header files currently supplied with the M JBILE software are summarized in tables 1 1 1 2 1 3 1 4 and 1 5 For ease of use header files are also summarized by groups in the container header files displayed in Table 1 6 By including one of these container header files the user includes automatically all of the header files of the corresponding groups This saves some typing leading to slightly longer compilation times although program size is not affected In order to access the objects of a module one must include the corresponding header file in the program Including a header file is accomplished by the directive MUIBILE 1 3 User s Guide 3 Section 1 6 Compiler issues include lt Mobile module name h gt Failure in including the correct header file will result in a large number of compiler errors such as CC Example C line 62 error MoElementaryJoint R1 MoElementaryJoint is not a type name 1314 CC Example C line 63 error MoElementaryJoint R2 MoElementaryJoint is not a type name 1314 CC Example C line 64 error MoElementaryJoint R3 MoElementaryJoint is not a type name 1314 As shipped from factory the MIJBILE package is organized in the following directories directory name description MOBILE_HOME_DIR Mobile header files MOBILE_HOME_DIR src implementation files not always available MOBILE
159. source code is typed in by the user is always generated automatically by the system as the project name with the extension cpp appended From the mobile model source file the executable M JBILE model is lateron automatically compiled and linked as a DLL Dynamic Link Library The files MoCmpnt cpp and MoCmpnt def are required for the interface between the main program MobileInventor exe and the executable MIJBILE model and should not be changed Open and edit the M JBILE model source file When opening the MIJBILE model source file you will encounter the following code include Mobile Inventor MoMFC h include Mobile Inventor MoScene h include Mobile Inventor MoWidget h _void main insert here your Mobile code return MoScene mainLoop Do not change any of these lines as otherwise the model will not compile and function correctly You may type in a MEXBILE model in the place indicated in the template much the same way as in the UNIX system The only difference is that all variables must MUIBILE 1 3 User s Guide Section 8 2 MI BILE for PC with Open Inventor be declared as static Not declaring all variables as static will result in loss of information and unpredictable errors An example of a program for PC is reproduced below include lt Mobile Inventor MoMFC h gt include lt Mobile MoElementaryJoint h gt include lt Mobile MoRigidLink h gt include lt Mobile MoMassElement h gt
160. surements involving different numbers of frames The topological types of measurements described in Table 5 6 and Fig 5 10 are applicable for the following tasks Absolute Motion Type I These objects project the absolute motion of a moving frame to a real number and subtract from it a constant y One can use these objects for example for establishing the height of an object over a coordinate plane or to compute the radius of a particle moving around a pole By the variable y which is mandatory one can establish an offset which is subtracted each time the measurement is carried out Such an offset can be for example of use for specifying an unloaded spring length for describing the constraint equation of a particle moving on the surface of a sphere etc Relative Motion Type II This is the classical topological measurement type It projects the relative motion of a reference frame Kpg with respect to another reference frame Kg to a scalar number As with type I a constant y has to be supplied that is subtrated from that number Objects of this type are useful for describing constraints arising from massless rods or couplers Also these objects serve as a basis for elementary force elements like springs dampers etc Absolute Difference Type III Objects of this kind compute a scalar num ber by subtracting the measurement obtained from the absolute motion of one mov ing frame kK from a corresponding measurement
161. t at a reference frame concrete MoLinearSpringDamper linear spring damper element for joints and chords concrete MoChord base class for geometric measurements abstract MoChordPointPointQuadratic measurement of quadratic distance between two origins concrete MoChordPointPointLinear measurement of linear distance between two origins concrete MoChordPointPlane measurement of distance between origin and plane concrete MoChordList list of chords concrete MoConstraintSolver base class for solution of loop constraint equations abstract MoExplicitConstraintSolver closed form solution of scalar constraint equations concrete MoImplicitConstraintSolver iterative solution of general constraint equations concrete MoMassElement inertia properties of point masses and rigid bodies concrete MoIntegrator base class for numerical integration abstract MoAdams Integrator Adams Moulton Bashfort integration concrete MoExplicitEulerIntegrator one step explicit Euler integration concrete MoRungeKuttaIntegrator 4th order Runge Kutta integration concrete MoDriver base class for objects generating kinematical input abstract MoConstantStepDriver generation of constant step increments for scalar variables concrete MoOutputGenerator printout of scalars vectors or matrices concrete Table 4 1 Selection of kinetostatic transmission elements in MiJBILE 4 2 1 Model of a Kinetostatic Transmission Element At the heart of the object oriented modeling of MIJBILE i
162. t checked for inconsistency For this reason it is not recommended to use the optimization parameters 78 MUJBILE 1 3 User s Guide Section 5 2 Basic Properties of Measurement Objects before a reference simulation has been carried out Ommitting all of optimization param eters will simply imply redundant calculations but no errors will be incurred Users not familiar with the multibody systems need not to be concerned with these issues The following code fragment illustrates a possible optimization setting for a linear point to point measurement of type IV MoFrame KB motion of this frame depends on input variable MoFrame KE motion of this frame depends on dependent variable MoFrame KBp KEp terminal frames of the upper segment MoVector d MoRigidLink coupler KBp KEp d upper segment MoChordPointPointLinear chord KEp coupler KE KB DO_BRANCH_II DO_BRANCH_I DO_BRANCH_II DO_BRANCH_I This constructor tells the measurement object chord that the dependent variable is situated somewhere below reference frame KE because of whereUnknown D0_BRANCH_I1 that the forces are requested for frames KB and KE because of whereForce D0_BRANCH_I DO_BRANCH_II and that input motion will only occur within the transmission chain below KB because of whereInput DO_BRANCH_I 5 2 8 Interlinking Measurements In the treatment of closed loops it may happen that the complete system of constraint
163. t possible to extend the provided li brary in any direction e Scalable approach treating all mechanical systems in a unified manner e Responsibility driven client server implementation simplifying the task of invoking the required functions and of implementing own costumized modules e Portable and efficient implementation based on the object oriented programming language C e Built in interfaces for three dimensional graphic libraries for animation with direct user feed back User interaction includes click and drag features for on line kine matics statics and dynamics this last feature may depend on system complexity and computer resources 1 2 Intended Audience This manual is addressed to users with a certain amount of experience with the modeling and simulation of mechanical systems Some familiarity with the C programming language is needed to understand and to apply the concepts described below However it is not necessary to master all of the many possibilities of the programming language C just to generate a MJJBILE model This is necessary only for developers planning to extend the MIJBILE package From the theoretical point of view some acquaintance is required with the basic concepts of kinematic and dynamic analysis of spatial mechanical systems This knowledge is only necessary at a very abstract level such as for deciding in which sequence a set of mechanical components needs to be traversed which kind of
164. th the kinetostatic state objects from one location of the system to the other Each transmission element supplies in analogy to its real world counterpart two basic operations I the transmission of motion and II the transmission of forces In MUBILE these two operations are realized as virtual functions doMotion and doForce respectively that are shared by all kinetostatic transmission elements Kinetostatic state objects serve as input and output variables for the various types of kinetostatic transmission elements There exist two basic types of kinetostatic state ob jects a spatial kinetostatic state objects or reference frames which can be imagined as interconnection junctures between pairs of kinetostatical transmission elements and 10 MUJBILE 1 3 User s Guide Section 2 1 Structure of MJJBILE b scalar kinetostatic state objects which represent actuator or sensor data used to drive the motors of the joints or to store scalar data extracted from the system by measurements The overall picture of the approach is illustrated in Fig 2 1 Prior to system assem bly reference systems are floating in space and possess no mutual relationship Scalar variables resemble wires waiting to be plugged into appropriate places of the kineto static transmission elements in order to generate the desired motion After assembly the reference systems become attached at specific points of the
165. that for the pure translation it does not matter with respect to which of the two frames K_Input and K_Output the relative displacement is decomposed The class MoRigidLink can also be employed for defining nary or multiple rigid links i e rigid links comprising n reference frames These links have one input reference frame and n 1 output reference frames which are collected in a reference frame list of type MoFrameList For each output reference frame a relative transformation matrix as well as a relative displacement vector can be defined which are collected in corresponding arrays Again displacement vectors can be defined with respect to the respective output frame or with respect to the input frame by placing the corresponding array after or before the array of rotation matrices respectively In the same way pure rotation or pure translation can be defined by leaving out the relative translations or relative rotations array respectively However it is only possible to make this selections equally for all output frames Hence if for example one output frame is only rotated and another only translated with respect to the input frame this can be taken advantage of only by introducing two separate rigid link objects The following example defines a binary link with its output frames located at the tips of a two links swivelled by 45 with respect to the input frame MoFrame K_Input K_Outputi K_Output2 MoRotationMatrix Delta_R 2
166. the example above the geometry is supplied automatically by the scene object However it is also possible to supply user defined geometries by specifying a corresponding data file as a second argument in makeShape The member function makeManipulator produces in addition to the graphical rendering a manipulator for the object in this case the joint through which the user can directly move the joint On a Silicon Graphics workstation this manipulation consists in dragging a cage around the joint Fig 2 3 shows the resulting graphics for the example above The graphical rendering is hardware dependent In M JBILE 1 3 the hardware supported are HP 9000 Series 700 Workstations and Silicon Graphics Indigo and Indy Workstations The display in Fig 2 3 stems from an SGI workstation The actual numerical simulation is carried out by an integrator of type Adams Bashfort Moulton After setting step size and error tolerance the integrator object can be treated as a kinetostatic transmission element that travels along the solution trajectory Each time the motion transmission function doMotion is invoked the system moves one small step along this trajectory After each such step the rendering function of the corresponding MUIBILE 1 3 User s Guide 17 Section 2 2 Example Analysis of a Simple Pendulum Figure 2 3 Graphic representation of the pendulum example graphical model of the pendulum is invoked bringing eventually the
167. tors was chosen as the modulus operator 4 which is the available operator most closely resembling the original mathematical symbol x Moreover this operator has precedence over the and operators so it conforms to common usage However the operator for dyadic product of two vectors chosen as has a lower precedence than the additive operators Thus expressions such as aob cod have to be programmed with additional levels of parenthesis i e as a b c7d for correct expression evaluation O subscripting pointer Lexpr function call expr expr_list complement expr unary minus expr unary plus t erpr multiply expr CXPr divide expr expr modulo remainder exprhexpr add plus expr expr subtract expr expr lt lt shift left expr lt lt expr gt gt shift right expr gt gt expr lt less than expr lt expr less than or equal expr lt expr greater than expr gt expr greater than or equal expr gt expr expr expr expr expr bitwise exclusive OR expr expr I logical inclusive OR expr expr simple assignment lualue expr multiply and assign lualue expr divide and assign lualue expr modulo and assign lualuef expr add and assign lualuet expr subtract and assign lualue expr AND and assign lualue amp expr inclusive OR and assign lvaluel expr exclusive OR and assign Ivalue expr p amp S oxsem Table 3 2 Precedence of operators in MJJBILE 20 MUJBILE 1 3
168. tostatic state objects 28 MUJBILE 1 3 User s Guide Section 3 4 State Objects state object state object transmission doMotion element kinematics 7 ES 7 aA TZ A statics RZ R_z Zz 1 doForce invocation read write read write user Figure 3 4 State objects as connectors between transmission elements b position velocit TES B acceleration Qg generalized force Figure 3 5 Structure of a scalar variable generalized coordinate e angular variables class MoAngularVariable e linear variables class MoLinearVariable These classes are derived from the generic class MoStateVariable which collects their common properties The class MoStateVariable is abstract i e it does not allow for direct instantiation of objects The only objects that can actually exist are those of type MoLinearVariable or MoAngularVariable However pointers to objects of type MoStateVariable are allowed The use of these pointers is explained further below The position velocity acceleration and force subentries of scalar kinetostatic state objects are accessed by apending q qd qdd or Q to the variable name respectively The type of the corresponding state subentry is summarized in Table 3 11 In some applications a discerning between the two types of scalar kinetostatic state objects is of no importance and one may wish to collect both types of kinetostatic state objects into one common list For e
169. transmission 00O transmit pre computed source forces no computation apply inertia related source forces applicable only to mass elements do all of the above Table 4 5 Possible values for the force subtask selection parameter For example a combination of position and velocity traversal is achieved by object doMotion DO_POSITION DO_VELOCITY 42 MUJBILE 1 3 User s Guide Section 4 3 Joints Links and Chains Theoretical background Calculation of Jacobians In MUSBILE 1 3 all calculations are based on a Jacobian free formulation This simplifies considerably the implementation of new classes However for some applications it may be required to determine the Jacobian Jy of a general mapping In the following two different methods are proposed for the calculation of Jacobians which make only use of the motion and force transmission functions A Velocity based determination of Jacobians column wise evaluation Setting at the input of the transmission element all velocity components equal to zero besides the jth one which is set to q 1 yields an output velocity vector which is identical to the jth column of the Jacobian _ J1 fri j i li 0 otherwise Joli 7 4 3 where Js denotes the jth column of Jy An equivalent approach is to use the acceleration transmission with the transmission substask selection parameter set to DO_EULER Force based determination of Jacobians row wise
170. transmission elements interconnecting them by pairs while the scalar variables accomplish the task of inducing motion at selected joints of the system The assembly of a mechanical system thus con sists in connecting the inputs and outputs of the kinetostatic transmission elements in appropriate order such that the resulting chains resemble the original system transmission elements actuator variables reference frames Figure 2 1 Objects in multibody systems The modeling of mechanical systems by kinetostatic transmission elements mirrors the client server paradigm of object oriented programming In this setting objects represent individuals that are endowed with specific responsibilities These responsibilities are chosen in such a way that the correct functioning of the overall society is warranted However the particular manner in which each object fulfills its responsibility is left as a matter of taste In MfXBILE the responsibilities of the mechanical elements are to provide the aforementioned virtual transmission functions For this functions it does not matter how an object realizes its task What matters is only that it does it MUIBILE 1 3 User s Guide 11 Section 2 2 Example Analysis of a Simple Pendulum 2 2 Example Analysis of a Simple Pendulum The following analysis of a simple mathematical pendulum shall illustrate the basic steps involved in the modeling of a mechanical system with MIJBILE The objective is to
171. ual values of the state variables in the array y The class MoDynamicSystem declares the overall behaviour of the object representing dif ferential equations in state space form For the actual generation of these equations dedicated classes have to be written In M ABILE there is currently only one class for doing this This class is termed MoMechanicalSystem and transforms a system of differ ential equations of second order resulting from the equations of motion of a mechanical system to state space form Other classes that are conceivable may generate first order equations for other types of systems such as hydraulic electric or control systems If needed these classes must be currently defined by the user By deriving them from the base class MoDynamicSystem the user can combine them with other objects represent ing state space form representations of dynamic equations and integrate them with the supplied numerical integrators 6 2 1 The Class MoMechanicalSystem Objects of class MoMechanicalSystem transform dynamic equations of mechanical systems into state space form Basically an object of type MoMechanicalSystem needs to be passed the name of a builder of equations of motion for generating the corresponding state space representation However Mf BILE provides also a shortcut that allows the user to avoid the need of constructing an extra object of type MoEqmBuilder This shortcut consists of passing to the object of type MoMechanicalS
172. var error constructor is private a compiler error will be issued because the constructor of the class MoStateVariable is not defined publicly A more elegant way of creating sets of heterogeneous scalar state variables is the use of variable lists Variable lists are created in MIJBILE as instances of class MoVariableList Variable lists support operations for appending additional items to the list moving for wards and backwards within the list and jumping to the beginning or the end of the list This functionality is realized through the operator lt lt and the functions getNext Q getPrevious rewind and jumpToEnd respectively Hereby the lists of variables adapt their storage requirements automatically so the user does not need to be concerned about implementation issues An example of the use of variable lists is the following code fragment Here the two scalar variables beta and s are collected in the variable list varlist from where they are retrieved further below in the program in order to print the value of their position subentry MoAngularVariable beta MoLinearVariable sg MoVariableList varlist varlist lt lt beta lt lt 8 beta q 90 0 RAD_TO_DEG s q 1 0 MoStateVariable p varlist rewind while p varlist getNext returns 0 at end of list MUIBILE 1 3 User s Guide 31 Section 3 4 State Objects switch p gt getType case
173. xample when generating the equations of motion or resolving constraint equations of complex mechanisms it does not matter whether the input coordinates are linear or angular In MJIBILE this can be done by introducing an array of pointers to their base class MoStateVariable and assigning to these pointers the values of the addresses of existing scalar kinetostatic state objects An example is the following code fragment in which the addresses of two scalar kinetostatic MUIBILE 1 3 User s Guide 29 Section 3 4 State Objects state access of type of entry subentry component angular case linear case position variable q MoAngle MoReal velocity vartable qd MoReal MoReal acceleration variable qdd MoReal MoReal force variable Q MoReal MoReal Table 3 11 Subentries for scalar kinetostatic state objects of type MoStateVariable state objects of different types namely beta and s are placed in the common array vars MoAngularVariable beta MoLinearVariable sg MoStateVariable vars 2 vars 0 amp beta vars i amp s After declaring this array of pointers one can access the state subentries of the concrete kinetostatic state objects through the pointer dereferencing mechanism For example the velocity acceleration and force components of both objects declared above can be accessed as follows vars 0 gt qd vars 0 gt qdd vars 0 gt Q vars 1 gt qd vars 1 gt qdd vars 1 gt Q 1 0 velocit
174. y introducing linear variables unnecessary evaluation of trigonometric expressions is avoided and the compiler can also make case selections based on the type of motion for example recognition of prismatic and revolute joints based on the type of the actuation variable TI BY linear scalars real line angular scalars real circle Figure 3 1 The two types of algebraic scalar objects in MIJBILE The type MoReal is just an alias of the native type double of C Thus all operations defined for the double precision variables are also defined for the type MoReal The type MoAngle groups together the value of an angle measured in radians and its sine and cosine The operations defined for this type are displayed in Table 3 4 An example of the use of angles is shown below include lt Mobile MoAngle h gt main MoAngle betal 0 beta2 beta3 betal is defined beta2 and beta3 not beta2 30 DEG_TO_RAD angle sine and cosine of beta2 are set 22 MUJBILE 1 3 User s Guide Section 3 3 Vectors and Matrices Operator usage Action angle angle angle real angle angle angle angle angle angle angle angle real angle angle angle real angle MoNullState angle degrees angle radians angle sine angle cosine angle assign an angle angle assign a scalar the value of the angle is in radians angle add two angles angle subtract two angles angle change sign
175. y subentries 2 0 acceleration subentries 3 0 force subentries For accessing the position it is necessary to know the exact type of kinetostatic state object addressed by the pointer as the type of this subentry depends on the type of the state variable This information is supplied by the member function getTypeQ It returns an object of type MoVariableType which can take on two values e PRISMATIC for linear variables and e REVOLUTE for angular variables The use of this function is illustrated in the following code fragment include lt Mobile MoVariableList h gt main MoAngularVariable beta MoLinearVariable sg MoStateVariable vars 2 vars 0 amp beta 30 MUJBILE 1 3 User s Guide Section 3 4 State Objects vars i amp s beta q 90 0 DEG_TO_RAD s q 1 0 for int i 0 i lt 2 i switch vars i gt getType case PRISMATIC cout lt lt MoLinearVariable vars i gt q lt lt n break case REVOLUTE cout lt lt MoAngularVariable vars i gt q lt lt n break The abstract pointer to MoStateVariable has been cast here to the correct type of con crete pointer before accessing the position subentry of the state variable Note that direct creation of objects of type MoStateVariable has been made impossible in MSXBILE by declaring the constructor as protected Thus if the user mistakenly types something like this MoStateVariable any
176. ystem the same arguments as to the object of MoEqmBuilder Thus there exist the following constructors for objects of type MoMechanical System MoMechanicalSystem MoEqmBuilder amp sys The object generates the state space form of the equations of motion established by the builder sys MoMechanicalSystem MoVariableList amp q MoMap amp phi MoMechanicalSystem MoVariableList amp q MoMapChain amp phiM MoMap amp phi MoMechanicalSystem MoVariableList amp q MoMap amp phi MoFrame amp K MoAxis zAxis 98 MUJBILE 1 3 User s Guide Section 6 3 Solving the Differential Equations MoMechanicalSystem MoVariableList amp q MoMapChaink phiM MoMap amp phi MoFrame amp K MoAxis zAxis MoMechanicalSystem MoVariableList amp q MoMap amp phi MoFrame amp K MoVector amp gravity MoMechanicalSystem MoVariableList amp q MoMapChain amp phiM MoMap amp phi MoFrame amp K MoVector amp gravity These objects generate the state space form of the dynamical equations of a me chanical system The meaning of the arguments is identical to that described in Section 6 1 6 3 Solving the Differential Equations Once the equations of motion are available in state space form they can be passed to a numerical integrator The numerical integrator is capable of moving along the solution trajectory of the system in prescribed time steps In this setting the integrator plays the role of a general

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