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Multibody System Simulation with SimMechanics Michael Schlotter May 2003 Abstract This paper describes, how to simulate the dynamics of multibody systems with SimMechanics, a toolbox for the Matlab/ Simulink environment. The novel Physical Modeling method is compared with traditional ways to represent mechanical systems in Matlab, and some interesting mathematical aspects of the implementation of SimMechanics are examined. 1 Introduction Simulating the dynamics of multibody systems is a common
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  Multibody System Simulationwith  SimMechanics Michael SchlotterMay 2003 Abstract This paper describes, how to simulate the dynamics of multibodysystems with  SimMechanics , a toolbox for the  Matlab/ Simulink  envi-ronment. The novel Physical Modeling method is compared with tra-ditional ways to represent mechanical systems in  Matlab , and some in-teresting mathematical aspects of the implementation of   SimMechanics are examined. 1 Introduction Simulating the dynamics of multibody systems is a common problem in engi-neering and science. Various programs are available for that task which areeither symbolical computation programs to derive and solve the dynamicalequations of motion, or numerical programs which compute the dynamics onthe basis of a 3D-CAD model or by means of a more abstract representation,e.g. a block diagram.As an add-on for the GUI-based simulation environment  Simulink ,  SimMe-chanics  falls into the last category. Mechanical systems are represented byconnected block diagrams. Unlike normal  Simulink  blocks, which representmathematical operations, or operate on signals,  Physical Modeling   blocks rep-resent physical components, and geometric and kinematic relationships di-rectly. This is not only more intuitive, it also saves the time and effort toderive the equations of motion. SimMechanics  models, however, can be interfaced seamlessly with ordinary Simulink  block diagrams. This enables the user to design e.g. the mechanicaland the control system in one common environment. Various analysis modes1  and advanced visualization tools make the simulation of complex dynamicalsystems possible even for users with a limited background in mechanics. 2 Functionality of the Toolbox This section provides an overview about  SimMechanics . The block set is de-scribed briefly, as well as the different analysis modes and visualization options.More details about these topics can be found in [Mat02]. 2.1 Physical Modeling Blocks As already mentioned, the  SimMechanics  blocks do not directly model math-ematical functions but have a definite physical (here: mechanical) meaning.The block set consists of block libraries for bodies, joints, sensors and actu-ators, constraints and drivers, and force elements. Besides simple standardblocks there are some blocks with advanced functionality available, which fa-cilitate the modeling of complex systems enormously. An example is the JointStiction Actuator with event handling for locking and unlocking of the joint.Modeling such a component in traditional ways can become quite difficult.Another feature are Disassembled Joints for closed loop systems. If a machinewith a closed topology is modeled with such joints, the user does not needto calculate valid initial states of the system by solving systems of nonlinearequations. The machine is assembled automatically at the beginning of thesimulation.All blocks are configurable by the user via graphical user interfaces as knownfrom  Simulink . The option to generate or change models from  Matlab  programswith certain commands is not implemented yet. It might be added in futurereleases. It is possible to extend the block library with custom blocks, if aproblem is not solvable with the provided blocks. These custom blocks cancontain other preconfigured blocks or standard  Simulink  S-functions.Standard  Simulink  blocks have distinct input and output ports. The con-nections between those blocks are called  signal lines  , and represent inputs toand outputs from the mathematical functions. Due to  Newton’s  third lawof actio and reactio, this concept is not sensible for mechanical systems. If a body  A  acts on a body  B  with a force  F ,  B  also acts on  A  with a force − F , so that there is no definite direction of the signal flow. Special  connection lines  , anchored at both ends to a  connector port   have been introduced withthis toolbox. Unlike signal lines, they cannot be branched, nor can they beconnected to standard blocks. To do the latter,  SimMechanics  provides Sen-sor and Actuator blocks. They are the interface to standard  Simulink  models.2  Actuator blocks transform input signals in motions, forces or torques. Sensorblocks do the opposite, they transform mechanical variables into signals. 2.2 Types of Analysis SimMechanics provides four modes for analyzing mechanical systems: Forward Dynamics  calculates the motion of the mechanism resulting fromthe applied forces/ torques and constraints. Inverse Dynamics  finds the forces/ torques necessary to produce a specifiedmotion for open loop systems. Kinematics  does the same for closed loop systems by including the extrainternal invisible constraints arising from those structures. Trimming  searches for steady or equilibrium states of a system’s motion withthe  Simulink  trim  command. It is mostly used to find a starting pointfor linearization analysis.Generally it is necessary to build a separate model for each type of analysis,because of the different mechanical variables which need to be specified. InForward Dynamics mode, the initial conditions for positions, velocities, andaccelerations, as well as all forces acting on the system are required to find thesolution. To use the Inverse Dynamics or the Kinematics mode, the model mustspecify completely the positions, velocities, and accelerations of the system’sindependent degrees of freedom. 2.3 Visualization Tools SimMechanics  offers two ways to visualize and animate machines. One is thebuild-in  Handle Graphics   tool, which uses the standard Handle Graphics fa-cilities known from  Matlab  with some special features unique to  SimMechanics .It can be used while building the model as a guide in the assembly process.If a body is added or changed in the block diagram, it is also automaticallyadded or changed in the figure window. This gives immediate feedback and isespecially helpful for new users or for complex systems. The visualization toolcan also be used to animate the motion of the system during simulation. Thiscan be much more expressive than ordinary plots of motion variables over time.The drawback is a considerably increased computation time if the animationfunctionality is used.The bodies of the machine can be represented as  convex hulls  . These are sur-faces of minimum area with convex curvature that pass through or surrounds3  all of the points fixed on a body. The visualization of an entire machine isthe set of the convex hulls of all its bodies. A second option is to representthe bodies as  equivalent ellipsoids  . These are unique ellipsoids centered atthe body’s center of gravity, with the same principal moments of inertia andprincipal axes as the body.More realistic renderings of bodies are possible, if the Virtual Reality Tool-box for  Matlab  is installed. Arbitrary virtual worlds can be designed with theVirtual Reality Modeling Language (VRML) and interfaced to the  SimMechan-ics  model. 3 An example problem Some of the features of   SimMechanics  will now be shown on an example. Theforward dynamics problem is solved with  Matlab/ Simulink  without the PhysicalModeling environment as well as with  SimMechanics . This allows a comparisonof the results and the effort taken to get the desired solution. It is also shownbriefly, how to compute the inverse dynamics and how to trim and linearizethe model with  SimMechanics . Finally, the problem of controlling an invertedpendulum is addressed as an example for the integrated design of mechanicaland control systems. 3.1 The Mechanism Figure 1 shows the mechanical system of interest. It consists of a uniformcubical rod  B  attached to a uniform cubical cart  A  which slides on a smoothhorizontal plane in reference frame  E  .  E   can be oriented arbitrarily in theNewtonian reference frame  N  .  A  and  B  are connected by means of a frictionlessrevolute joint with the axis of rotation in  A 3  direction.  A  is attached to  E  with a linear spring/ damper system with the spring stiffness  k SD   and thedamping coefficient  b SD  . The dimensions of the cart in  A 1 ,  A 2 ,  A 3  directionare  a A ,  b A ,  c A , respectively, and the dimensions of the rod are similarly definedin reference frame  B  as  a B ,  b B ,  c B . The masses and central inertia dyadics of bodies  A  and  B  are assumed to be  m A ,  I A ,  m B , and  I B . The gravity forceacts in  − N 3  direction. Two spatial forces,  F A ∗ and  F P  1 act on the mechanismas shown. The system has one translational degree of freedom,  q  1  and onerotational degree of freedom  q  2 .The equations of motion for that mechanism can be derived by hand. They4
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