Simulation of Mechanisms-
Analytical and Web-based
Virtual Prototyping: A
Comparative Study
Under the Supervision of Dr. K...
Simulation of Mechanisms: Analytical and Web-based Virtual
             Prototyping- A Comparative Study
                 ...
Contents
Introduction ………………………………………………………………………..3

Objective …………………………………………………………………………...3

Case Study 1: Simple Pen...
Introduction
        In the recent years there has been tremendous advancement in computer technology. The
engineering dom...
1) VisualNastran
   2) SimMechanics
   3) MapleSim

Implementation
                           CREATE THE MODEL IN SOLIDWOR...
Case Study 1: Simple Pendulum
Analytical Approach




                               FIGURE 1: Simple Pendulum

Mass of th...
cos 
                                              = −
                                                            2
Step4...
FIGURE 2: Isometric view of the link and the pin

The length of the link (pendulum) is set to 1 meter and the breadth as 0...
1. VisualNastran Menu appears in SolidWorks. Launch VisualNastran by clicking the
      connect button in the VisualNastra...
FIGURE 5: Constraint Navigator

In the steps to follow, you will adjust the joints to give each the appropriate degrees of...
FIGURE 6: Isolated Constraint

       2. Click the Move button in the Constraint Navigator, click over the pendulum, and d...
The first step in simulating the motion of the piston assembly is to make sure it is properly
anchored.

When VisualNastra...
3. Click the Motor tab.
      This displays the motor page of the properties window.
   4. You can select either of the fo...
We simulate the pendulum under gravity setting the initial angle to be pi/3. And meter the
angular orientation, velocity a...
4. We get an icon which indicates that we imported the model from SolidWorks.




             FIGURE 9: Icon of SolidWork...
FIGURE 11: Configuring Parameters in SimMechanics

7. Then select Update Diagram from Edit menu, which shows the mechanism...
FIGURE 12: Simple Pendulum in SimMechanics Virtual Environment

      8. The gravity settings and Analysis mode (Forward D...
4. And we can monitor the output by connecting the joint sensor to a scope.

       The final Simulink Block diagram looks...
FIGURE 14: Angular units, Velocity and Acceleration of the Pendulum

         Similarly we can simulate the reaction force...
Case Study 2: Two Link Mechanism
Analytical Approach




                     FIGURE 15: Two Link Mechanism



           ...
1        
                                      =        ,  = 1
                                             2        2

S...
1       1
                                          =        , = 
                                                 2      ...
Virtual Simulation
Step 1: Creating the model in SolidWorks




        FIGURE 16: Front View of a Two Link Mechanism in S...
FIGURE 17: Isometric View of the Two Link Mechanism in VisualNastran

The motion of the two link mechanism is simulated un...
FIGURE 18: SolidWorks Model




           FIGURE 19: Simulink Model with Virtual Reality Toolbox Animation

   To conver...
FIGURE 20: Creating an XML File




FIGURE 21: Importing to SimMechanics


              26
FIGURE 22: Icon of SolidWorks model in Simulink




                      FIGURE 23: SimMechanics Model

Go to Simulation...
FIGURE 24: Configuring Parameters in SimMechanics




                           FIGURE 25: Animation Window

Not a very r...
    Set the level of detail in SolidWorks
1.   VRML 97 output format
2.   Separate files
3.   Units-meters
    Set the E...
FIGURE 27: Save as VRML




FIGURE 28: Settings to be used while saving




                 30
Process3: Prepare the virtual scene

      Correct the VRML export filter shortcoming
      Create an aggregated main fi...
FIGURE 30: Notice the ShortComing




FIGURE 31: Correct the Shortcoming and save




                 32
To add transform nodes and do other settings open the VRMl file with VREALM builder




                             FIGUR...
FIGURE 34: Adding Body Sensor Blocks




                                FIGURE 35: Add GOTO blocks

    Process5: Add a V...
FIGURE 36: Creating a subsystem




FIGURE 37: Add FROM block and VR Sink block to subsystem


                        35
FIGURE 38: Appropriate settings by double-clicking VR Sink block




    FIGURE 39: Add the appropriate tags to the FROM b...
FIGURE 40: Use Rotation matrix to VR




     FIGURE 41: Virtual Model


             37
Customizing our model to perform the required task:
The task here is to trace an ellipse. So we need to actuate the revolu...
FIGURE 43: SubSystem of the Simulink model

Clicking the simulation, we could see that the two-link manipulator traces the...
FIGURE 44: Error in joint angle one in SimMechanics




FIGURE 45: Error in joint angle 2 using SimMechanics


           ...
FIGURE 46: Error in joint angle one using Analytical method




 FIGURE 47: Error in joint angle 2 using Analytical method...
Analytical                                   SimMechanics

                    10                                         ...
Case Study 3: Four Bar Mechanism
Analytical Approach




                                  FIGURE 48: Four Bar Mechanism

...
3 = (l1 sin θ1 θ1 + l2 sin θ2 θ2 )/(l3  3 )

      Let us drive this model by giving a constant velocity to the crank.

  ...
FIGURE 50: Four Bar Mechanism in VisualNastran Environment



Step3: Exporting the Four Bar to SimMechanics:

The Four Bar...
FIGURE 51: SimMechanics Model




FIGURE 52: Final SimMechanics Model with Virtual Reality Animation




                 ...
Results

                   Analytical         SimMechanics    VisualNastran

          Inputs    Outputs

      1 = 90   ...
Case Study 4: Wheeled Mobile Robot
Dynamic Analysis
Consider the Wheeled Mobile Robot as shown in the figure below:

The f...
:           

:      

 :              

       DC motors

 :            

  :             

               
             ...
FIGURE 54: Notation for the geometry of the mobile platform

Where

                              − sin ∅     cos ∅     − ...
+  ,  +   = 

Where,

                                          0                        2  sin ∅        0        0
      ...
The Corresponding decoupling matrix for the output is:

                                                          ∅11    ∅...
FIGURE 55: Simulink Model of WMR

Results:

The look ahead point should trace an ellipse of given parameters. The motion o...
FIGURE 57:  and  vs time




FIGURE 58:  and  vs time




              54
FIGURE 59: ∅ vs time




         55
Symbolic Modeling using MapleSim
       MapleSim is a complete environment for modeling and simulating complex multidomain...
FIGURE 60: MapleSim Model of Simple Pendulum

Obtaining the multibody equations and solving them

The multibody equations ...
Solution of EOM




Results




                  FIGURE 61:  vs time


                          58
FIGURE 62: Animation of the Pendulum



Initial Condition:

 = pi/4 from vertical


                                      ...
Results

Simulating the system for 10 secs in Matlab and in MapleSim gives us the following results:




          FIGURE ...
FIGURE 65:  in Matlab and in MapleSim




                  61
Case Study 2: Two Link Mechanism
To simulate the motion of a double pendulum under gravity in Matlab and MapleSIm and
comp...
63
Equations of Motion




                      64
Results




          65
FIGURE 67:  vs time




          66
FIGURE 68:  vs time




          67
FIGURE 69: Animation of Two link mechanism




Initial conditions:

1 = -pi/4;

2 = -pi/4;

Simulation Time = 10 secs.



...
FIGURE 70: X-Y Position of the End-Effector in Matlab and MapleSim




                      FIGURE 71:  vs time

        ...
FIGURE 72:  vs time




FIGURE 73:  vs time




          70
FIGURE 74:  vs time




          71
Case Study 3: Four-Bar Mechanism
Dynamic Analysis




                                FIGURE 68: A Fourbar Mechanism



  ...
The Equations of Motion(EOM) for the overall system are derived by treating the four-bar
linkage as being composed of two ...
For Dyad 2




The constraint equations are then obtained from the requirement that Dyad 1 and Dyad 2 to stay
connected at...
Where




From the constraint equation we can get




We can also determine as




That is



It means that the above expr...
EOM can be rewritten to




We can show that,




We can also derive that




We multiply ST to both sides of EOM ,




In...
MapleSim Model

The model looks as shown in the figure below




                          FIGURE 70: MapleSim Model of a ...
Equations of Motion (EOM)




                            78
79
Results




          FIGURE 71: X-Y trajectory of the point of interest in Matlab and MapleSim




                      ...
FIGURE 73:  vs time




FIGURE 74:  vs time




          81
Conclusion
    Thus, we can clearly see that the analytical formulation of the above examples is a very
    tedious and ti...
References

     “Control of Mechanical Systems with Rolling Contraints: Application to Dynamic
      Control of Mobile R...
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Virtual Prototyping Project - PDF

  1. 1. Simulation of Mechanisms- Analytical and Web-based Virtual Prototyping: A Comparative Study Under the Supervision of Dr. Krovi Sasi Bhushan Beera 1
  2. 2. Simulation of Mechanisms: Analytical and Web-based Virtual Prototyping- A Comparative Study by Sasi Bhushan Beera Under the guidance of Dr. Krovi 2
  3. 3. Contents Introduction ………………………………………………………………………..3 Objective …………………………………………………………………………...3 Case Study 1: Simple Pendulum ……………..……………………………………5 Case Study 2: Double Pendulum ……………..…………………………………..19 Case Study 3: Four Bar Mechanism ……………..…………………………….....41 Case Study 4: Wheeled Mobile Robot ……………..…………………………….47 Symbolic Modeling using MapleSim……………………………………………..55 Conclusion………………………………………………………………………...67 References………………………………………………………………………...68 3
  4. 4. Introduction In the recent years there has been tremendous advancement in computer technology. The engineering domain had benefited significantly with the application of computational power in many fields like design, development, manufacturing and analysis etc. In the case of mechanisms Mathematical/Analytical models are generally used to represent simple models. Such models can be analyzed by developing the equations of motion and solving them analytically. These models can be used as a starting step in the analysis process. However, modeling complex geometries analytically is always a herculean task. With the advent of modeling techniques like Virtual Prototyping (VP) modeling and analysis of complex geometries becomes much easier and saves a lot of time. Computer based simulation tools can now be used to model mechanisms and calculate their kinematic and dynamic responses which can be visualized in a 3-D virtual environment. Some of the advantages of these techniques are (a) Simplicity in Modeling (b) Ability to accelerate the process. However, it must be always kept in mind that these tools work on the principle “Garbage in, garbage out”. Objective The main objective of this study is to compare the various simulation tools from the view point of: 1) Simplicity in Modeling 2) Accuracy of Modeling As discussed in the previous section equations of motion for simple mechanisms are developed and solved analytically. And then for the same mechanisms simulation tools are used for modeling and analysis. The results are then compared. Analytical Approach  Idealize by making assumptions  Drawing Free Body Diagrams  Formulating EOM‟s  Solving for them Simulation Based Design With the development of computational technology, a number of simulation tools have been developed for modeling complex mechanical systems. Some of the tools which we will use to accomplish the objective of our study are listed below: 4
  5. 5. 1) VisualNastran 2) SimMechanics 3) MapleSim Implementation CREATE THE MODEL IN SOLIDWORKS EXPORT THE MODEL TO EXPORT THE MODEL TO SIMMECHANICS VISUALNASTRAN SIMULATE AND VISUALIZE THE MOTION PLOT AND ANALYZE THE RESULTS The various examples considered are: 1) Simple Pendulum 2) Two Link 3) Four Bar 4) Wheeled Mobile Robot 5
  6. 6. Case Study 1: Simple Pendulum Analytical Approach FIGURE 1: Simple Pendulum Mass of the Link m = 2Kg Length of the Link l = 2m TABLE 1 Lagrangian Formulation Step1: Introduce generalized coordinates and generalized speeds = , = Step2: Define Kinetic Energy (T) 1 1 = 2 + 2 2 2 Step3: Define Potential Energy (V) 6
  7. 7. cos = − 2 Step4: Power (π) = = 0 Step5: Dissipation (∆) 1 ∆= 2 2 Step6: Lagrangian (L) = − Step7: Equation of Motion (EOM) ∆ : − = − Substituting the above defined terms in EOM we get the generalized equation of motion as: 1 2 1 + sin = 0 3 2 Task: To give an initial joint angle of 30 deg and simulate its motion.s Virtual Simulation We use SolidWorks to model the simple pendulum and then export it to either VisualNastran or SimMechanics to simulate its motion. The basic steps of modeling and simulation are shown in the flow diagram below: 1. Creating the model in SolidWorks: Create the model of the simple pendulum in SolidWorks which basically consists of two parts, the pin and the link. The pin is the fixed part and the link is the part that rotates like a pendulum and both of them are connected by a revolute joint. The model is shown below: 7
  8. 8. FIGURE 2: Isometric view of the link and the pin The length of the link (pendulum) is set to 1 meter and the breadth as 0.1 meter. The dimensions of the above assembly can be seen in the figure above. 2. Exporting the model to VisualNastran and simulating its motion: The following steps are carried out in this process:  Exporting the model  Simulating its motion  Plot and analyzing the results Exporting the model to VisualNastran: To activate MSC.VisualNastran desktop for SolidWorks, We need to enable it using the SolidWorks Add-Ins feature.  Launch the SolidWorks program  Choose Add-Ins in SolidWorks tools menu  Check the VisualNastran checkbox 8
  9. 9. 1. VisualNastran Menu appears in SolidWorks. Launch VisualNastran by clicking the connect button in the VisualNastran menu. FIGURE 3: VisualNastran menu in SolidWorks When the export is complete, the MSC.VisualNastran Desktop program opens and displays the CAD Associativity dialog, listing the MSC.visualNastran Desktop objects that are associated with the objects in the SolidWorks model as shown below: FIGURE 4: CAD Associativity Dialog 2. Click OK to close the CAD Associativity dialog. MSC.VisualNastran Desktop prompts you to run the constraint Navigator. 3. Click Yes to run the Constraint Navigator. The Constraint Navigator allows you to examine the relationships among bodies, subassemblies, and constraints so that you can verify and modify your simulation model. 9
  10. 10. FIGURE 5: Constraint Navigator In the steps to follow, you will adjust the joints to give each the appropriate degrees of freedom. 1. Click the next constraint button. Concentric1, a revolute constraint, is isolated as shown in figure below: 10
  11. 11. FIGURE 6: Isolated Constraint 2. Click the Move button in the Constraint Navigator, click over the pendulum, and drag the mouse. This tests the kind of movement allowed by the constraint. This constraint is OK. 3. Similarly check other constraints. In our case we have only one revolute constraint. 4. Click Done to close the Constraint Navigator. Anchoring Assembly: If a body is “fixed” in SolidWorks, it will be anchored in MSC.VisualNastran Desktop. And by default, SolidWorks “fixes” the first body brought into an assembly. In our model the pin is fixed and the link can rotate about the pin which is exactly what we want. Simulating the Motion: 11
  12. 12. The first step in simulating the motion of the piston assembly is to make sure it is properly anchored. When VisualNastran imports the body from SolidWorks, it anchors it appropriately. No modifications are necessary. You are ready to run an initial simulation. 1. Click the Run button in the Tape Player Control, and let the simulation run. VisualNastran begins to simulate the motion of the model. The geometry of the assembly causes the link to swing like a pendulum around the pin. Since this is the first time the simulation is being run, VisualNastran calculates the dynamics and stores the data. 2. Repeat the simulation by clicking the Stop button, then the Reset button, and then the Run button again. Depending on the speed of the computer, the animation may be faster this time because the history has already been calculated. Adding a Motor: Currently, the assembly is moving only in response to the effects of gravity. In this step, you will change the joint between the pin and the link to a motor that drives the motion of the system. 1. Right click on concentric1 and select properties. 2. Then select Revolute Motor from the list of available joint types as shown below. The revolute joint is changed to revolute motor. FIGURE 7: Properties Window for Revolute Motor 12
  13. 13. 3. Click the Motor tab. This displays the motor page of the properties window. 4. You can select either of the four: Orientation, Angular velocity, Angular Acceleration and Torque. Enter a value and simulate the pendulum. Results: The position, velocity, acceleration of the link and constraint torques and joint forces can be computed at each point in time as simulation takes place. 1. Select Meter from the Insert button. And from Meter you can select orientation, velocity, acceleration, torque or joint forces to be mapped as we run the simulation. FIGURE 8: Results shown at the bottom of the window Now, we would want to compare the results obtained by VisualNastran with analytical results. 13
  14. 14. We simulate the pendulum under gravity setting the initial angle to be pi/3. And meter the angular orientation, velocity and acceleration as described above. For simplicity, find the joint angles, rates and accelerations at three different times and compare them with the analytical results which can be obtained by the EOM of the Simple Pendulum given above. The results are tabulated as shown below: Analytical VisualNastran Error θ θ -23.1 -330.445 -351 0.062203998 10.9 159.265 169 0.061124541 3.81 55.966 59.4 0.061358682 TABLE 2 3. Exporting to SimMechanics : The following steps are carried out in this process:  Exporting the model  Simulating its motion  Plotting and analyzing the results Exporting the model to SimMechanics:  Launch the SolidWorks program  Choose Add-Ins in SolidWorks tools menu  Check the SimMechanics checkbox. 1. SimMechanics menu appears in SolidWorks. Click settings from the menu and leave the settings default. Then save the file as .xml 2. Open Matlab and go to the folder that has the .xml file 3. Then type the command “import_physmod(„filename.xml‟). It takes the xml file and converts the SolidWorks model into a Simulink block diagram. 14
  15. 15. 4. We get an icon which indicates that we imported the model from SolidWorks. FIGURE 9: Icon of SolidWorks model in Simulink 5. Double click the icon to get the whole mechanism represented in terms of Simulink block diagram. FIGURE 10: Simulink Block Diagram 6. Enable the inherent virtual environment window in SimMechanics. Go to Simulation, Configuration parameters and select SimMechanics and check the two dialog boxes as shown in the figure below. 15
  16. 16. FIGURE 11: Configuring Parameters in SimMechanics 7. Then select Update Diagram from Edit menu, which shows the mechanism in its own virtual environment. It may not be very clear like SolidWorks since it takes mass properties of the elements into account. 16
  17. 17. FIGURE 12: Simple Pendulum in SimMechanics Virtual Environment 8. The gravity settings and Analysis mode (Forward Dynamics, Inverse Dynamics etc) can be selected by clicking the Environment Block. 9. The time settings can be selected by clicking Configuration Parameters, Solver and entering the values for time settings. Case1: Simulation under gravity Case2: Simulation with a motor 1. We can actuate the revolute joint by clicking the revolute joint icon in the Simulink Block diagram and enter the number of sensor and actuator ports. 2. The sensor and actuator ports can be connected to a sensor and actuator block respectively by which we can actuate the joint and measure the various parameters. 3. We can actuate the joint either by torque or by motion. Lets actuate it with motion and give a sinusoidal input. 17
  18. 18. 4. And we can monitor the output by connecting the joint sensor to a scope. The final Simulink Block diagram looks like FIGURE 13: Final Simulink Block Diagram Results: The simple pendulum is simulated under gravity and with the motor actuated by a sinusoidal input. And we could monitor the angular units, angular velocity, angular acceleration, reaction torque and reaction force at each instant in time. 18
  19. 19. FIGURE 14: Angular units, Velocity and Acceleration of the Pendulum Similarly we can simulate the reaction forces and torques on the pendulum. Comparison: Analytical SimMechanics θ θ Error 23.76 -339.347 -356.758 0.051307364 17.44 -252.427 -264.723 0.048711113 -3.96 58.165 63.506 0.091824981 -27.57 389.819 413.717 0.061305375 TABLE 3 19
  20. 20. Case Study 2: Two Link Mechanism Analytical Approach FIGURE 15: Two Link Mechanism Link Lengths 1 = 2, 2 = 1 Distance of Mass Centers = , = 1,2 2 Link Masses and Inertias 1 = 10, 2 = 6, 1 = 3, 2 = 2 TABLE 4 Lagrangian Formulation We follow the seven step procedure demonstrated for the case of Simple Pendulum to find the equations of motion for the two link mechanism. Step1: Introduce generalized coordinates and generalized speeds 20
  21. 21. 1 = , = 1 2 2 Step2: Define Kinetic Energy (T) 1 1 = 2 + 2 2 , = 1,2 2 2 = 1 + 2 We get Kinetic energy as, 1 1 1 1 1 = 1 2 + 1 + 2 2 1 + 2 2 + 2 2 1 1 2 2 2 2 2 2 2 2 + 22 1 2 1 2 cos 2 − 1 Step3: Define Potential Energy (V) = 1 + 2 We get Potential Energy as, = (1 + 2 )1 + 1 1 + 2 1 sin 1 + 2 2 sin 2 Step4: Power (π) = 1 1 + 2 (2 − 1 ) Step5: Dissipation (∆) 1 2 1 2 2 ∆= 1 1 + 2 (2 − 1 ) 2 2 Step6: Lagrangian (L) = − Step7: Equation of Motion (EOM) ∆ : − = − Substituting the above defined terms in we get the generalized equation of motion as + , + = 21
  22. 22. 1 1 = , = 2 2 2 1 2 2 1 + 1 2 1 2 1 cos(2 − 1 ) = 4 2 2 2 2 1 cos(2 − 1 ) 2 + 2 2 /4 2 l2 2 −m2 l1 sin θ2 − θ1 2 2 = l2 2 −m2 l1 sin θ2 − θ1 1 2 1 1 cos θ1 + 2 1 cos 1 = 2 2 2 cos 2 2 Task: To trace an ellipse Task To trace an ellipse Center = 0, = 0 Semi major and Minor Axes 1 = 1.75, 2 = 1.25 Ellipse Orientation = /4 Angular Velocity = 72/180 TABLE 5 Controls Used: Kinematics Closed loop Task Space Control Dynamics Closed loop Joint Space Control 22
  23. 23. Virtual Simulation Step 1: Creating the model in SolidWorks FIGURE 16: Front View of a Two Link Mechanism in SolidWorks Step2: Exporting the Two link mechanism to VisualNastran The two link mechanism developed in SolidWorks is then exported to VisualNastran to simulate its motion. 23
  24. 24. FIGURE 17: Isometric View of the Two Link Mechanism in VisualNastran The motion of the two link mechanism is simulated under gravity and by actuating its joints. The joints can be actuated by a revolute motor and velocity, acceleration and torque profiles can be given as inputs to simulate the motion. Step3: From SolidWorks Assemblies to Simmechanics Models and Virtual Reality Toolbox Animations The two link mechanism is then exported to SimMechanics. Now let us have our end effector trace the trajectory of a circle. The joint angles, rates and accelerations are computed analytically and they are given as inputs to the respective joints to obtain the required task space motion. 24
  25. 25. FIGURE 18: SolidWorks Model FIGURE 19: Simulink Model with Virtual Reality Toolbox Animation  To convert the SolidWorks assembly into a SimMechanics model  To create an animation of the model: -Export the SolidWorks assembly into a virtual world and link it to the SImMechanics model using Virtual Reality Toolbox Process1: Convert the SolidWorks Assembly into a SimMechanics model  Save the SolidWorks assembly into an XML file  Create a SimMechanics model 25
  26. 26. FIGURE 20: Creating an XML File FIGURE 21: Importing to SimMechanics 26
  27. 27. FIGURE 22: Icon of SolidWorks model in Simulink FIGURE 23: SimMechanics Model Go to Simulation Configuration parameters SImMechanics Show Animation and Simulation 27
  28. 28. FIGURE 24: Configuring Parameters in SimMechanics FIGURE 25: Animation Window Not a very realistic animation. We want to connect Virtual Reality Toolbox to get a realistic animation. Process2: Export the SolidWorks Assembly as a VRML file 28
  29. 29.  Set the level of detail in SolidWorks 1. VRML 97 output format 2. Separate files 3. Units-meters  Set the Export Options  Export FIGURE 26: Setting image quality 29
  30. 30. FIGURE 27: Save as VRML FIGURE 28: Settings to be used while saving 30
  31. 31. Process3: Prepare the virtual scene  Correct the VRML export filter shortcoming  Create an aggregated main file  Add Transform nodes  Add - World information - Navigation information - Lighting - Top-level Transform node - View points FIGURE 29: Open the VRML file as text file 31
  32. 32. FIGURE 30: Notice the ShortComing FIGURE 31: Correct the Shortcoming and save 32
  33. 33. To add transform nodes and do other settings open the VRMl file with VREALM builder FIGURE 32: VREALM BUILDER Process4: Create Signals to connect to a virtual scene  For each body, add a unity coordinate system  Add Body Sensor blocks: - Measure position and rotation matrix - Set units to meters  Add Goto and From blocks for keeping the model clean FIGURE 33: Creating Unity coordinate system 33
  34. 34. FIGURE 34: Adding Body Sensor Blocks FIGURE 35: Add GOTO blocks Process5: Add a Virtual Scene to the model  Add the VR Sink block to the model  For each body in the virtual scene, select translation and rotation  Use Rotation Matrix to the VRML Rotation block 34
  35. 35. FIGURE 36: Creating a subsystem FIGURE 37: Add FROM block and VR Sink block to subsystem 35
  36. 36. FIGURE 38: Appropriate settings by double-clicking VR Sink block FIGURE 39: Add the appropriate tags to the FROM blocks 36
  37. 37. FIGURE 40: Use Rotation matrix to VR FIGURE 41: Virtual Model 37
  38. 38. Customizing our model to perform the required task: The task here is to trace an ellipse. So we need to actuate the revolute joints accordingly. For this case let us do kinematic actuation. We will explore dynamic actuation in our next example, Four-bar mechanism. Kinematic actuation implies we simply actuate the motors at both the revolute joints using motion (joint angles, rates and acceleration). From inverse kinematic analysis we can come up with equations for joint rates and pass them through integrator and differentiator blocks to find joint angles and joint accelerations respectively. The final model in Simulink is as shown below. FIGURE 42: Simulink block diagram of final model 38
  39. 39. FIGURE 43: SubSystem of the Simulink model Clicking the simulation, we could see that the two-link manipulator traces the desired trajectory. Comparison with Analytical Solution: However, our goal here is to compare the SimMechanics solution with the analytical solution that we have from Matlab. The criteria of comparison here is the error in joint angles. Error is equal to desired joint angles minus the actual joint angles. The results are shown in the figures below. 39
  40. 40. FIGURE 44: Error in joint angle one in SimMechanics FIGURE 45: Error in joint angle 2 using SimMechanics 40
  41. 41. FIGURE 46: Error in joint angle one using Analytical method FIGURE 47: Error in joint angle 2 using Analytical method 41
  42. 42. Analytical SimMechanics 10 13 10 12.5 TABLE 5: Maximum errors in joint angles Comparing the above to graphs, we could clearly see that the errors are less in Analytical method as compared to SimMechanics. However, the difference is not so much. So taking into account the ease of modeling and animation into criteria we could clearly deploy SimMechanics for Virtual Prototyping. 42
  43. 43. Case Study 3: Four Bar Mechanism Analytical Approach FIGURE 48: Four Bar Mechanism Link Lengths 0 = 1, 1 = 1, 2 = 4, 3 = 1.5 TABLE 6 Kinematic Analysis Loop Closure Equations 1 1 + 2 2 = 0 + 3 3 1 1 + 2 2 = 3 3 We can differentiate the above two equations with respect to time to come up with velocity level equations: −1 sin 1 1 − 2 sin 2 2 = −3 sin 3 3 1 cos 1 1 + 2 cos 2 2 = 3 cos 3 3 Given the angular velocity of crank: 1 = w2 = p/6 rad/s, we can compute the angular velocities of the other two links by solving the above two equations: θ2 = (−1 sin 1 + 1 1 tan 3 1 )/( −1 cos 2 tan 3 + 2 sin 2 ) 43
  44. 44. 3 = (l1 sin θ1 θ1 + l2 sin θ2 θ2 )/(l3 3 ) Let us drive this model by giving a constant velocity to the crank. Virtual Simulation Step 1: Four Bar Mechanism in SolidWorks FIGURE 49: Isometric View of Four Bar in SolidWorks Step2: Exporting the Four Bar to VisualNastran The Four Bar mechanism can be exported to VisualNastran just like in the case of Simple Pendulum. The model looks as shown in the figure below in the VisualNastran environment. 44
  45. 45. FIGURE 50: Four Bar Mechanism in VisualNastran Environment Step3: Exporting the Four Bar to SimMechanics: The Four Bar mechanism is then exported to SimMechanics and here we can actuate the revolute joints with the required inputs. And we also connect it to Virtual world for much better animation. 45
  46. 46. FIGURE 51: SimMechanics Model FIGURE 52: Final SimMechanics Model with Virtual Reality Animation 46
  47. 47. Results Analytical SimMechanics VisualNastran Inputs Outputs 1 = 90 2 = −26.9 2 = −27.23 2 = −27.2 2 = 20 3 = 13.42 3 = 13.5136 3 = 13.5 3 = 72.9 1 = 30 1 = 137.4 2 = −24.4 2 = −24.6 2 = −24.59 2 = 20.8 3 = 11.9 3 = 11.7 3 = 11.64 3 = 93.4 1 = 30 TABLE 7 47
  48. 48. Case Study 4: Wheeled Mobile Robot Dynamic Analysis Consider the Wheeled Mobile Robot as shown in the figure below: The following notation will be used in the rest of our discussion: − : − : the coordinate system fixed to the cart : , : , : ( , ) FIGURE 53: Wheeled Mobile Robot Where, 48
  49. 49. : : : DC motors : : : : : : − If we ignore the passive wheels, the configuration of the platform can be described by five generalized coordinates. These are the three variables that describe the position and orientation of the platform and two variables that specify the angular positions for the driving wheels. Therefore, let = xc , yc , ∅, θr , θl Where ( , ) are the coordinates of center of mass in the world coordinate system and ∅ is the heading angle of the platform as shown in FIGURE 50, θr and θl are the angular positions of right and left driving wheels respectively. Assuming the driving wheels roll (and do not slip) there are three constraints. cos ∅ − sin ∅ − ∅ = 0 cos ∅ + sin ∅ + ∅ = cos ∅ + sin ∅ − ∅ = The three constraints can be written in the form: = 0 49
  50. 50. FIGURE 54: Notation for the geometry of the mobile platform Where − sin ∅ cos ∅ − 0 0 − cos ∅ sin ∅ − 0 − cos ∅ − sin ∅ 0 Thus the mechanical system has two degrees of freedom. It is straightforward to verify that the following matrix c(b cos ∅ − d sin ∅) c(b cos ∅ + d sin ∅) c(b sin ∅ + d cos ∅) c(b sin ∅ − d cos ∅) S q = c −c 1 0 0 1 Which satisfies () = 0, where the constant = /2. We now derive the dynamic equation for the mobile platform. The lagrange equations of motion of the platform are given by 50
  51. 51. + , + = Where, 0 2 sin ∅ 0 0 0 −2 cos ∅ 0 0 = 2 sin ∅ −2 cos ∅ 0 0 0 0 0 0 0 0 0 0 2 ∅2 cos ∅ 0 0 2 ∅2 sin ∅ 0 0 , = 0 = 0 0 = 0 1 0 0 0 1 1 θ = = l 2 θr The task here is such that the look ahead point should trace a given trajectory. Now is given by the following relation: L1 = T L2 Where cos ∅ − sin ∅ xc = sin ∅ cos ∅ yc 0 0 1 Since the system has two inputs, we may choose any two output signals, we may choose any two output variables. We consider the following four types of output equations: Type I: = = , Type II: = = , ∅ Type III: = = , ∅ Type IV: = = 1 , 2 () The Type I output equation results in a trajectory tracking control system which has been studied. 51
  52. 52. The Corresponding decoupling matrix for the output is: ∅11 ∅12 Ф= ∅21 ∅22 ∅11 =c b − L2 cos ∅ − (d + L1 ) sin ∅ ∅12 =c b + L2 cos ∅ + (d + L1 ) sin ∅ ∅21 =c b − L2 sin ∅ + (d + L1 ) cos ∅ ∅22 =c b + L2 sin ∅ − (d + L1 ) cos ∅ Ф Ф ∅ = ∅= (1 − 2 ) ∅ ∅ = ( ) + + u = ∅−1 q − ∅ 1 = 1 2 = 2 Simulink Integration The above dynamic formulation is used to prepare a Simulink block diagram and connect it to VR toolbox and simulate the motion of a WMR. The block diagram is as shown In the figure below: 52
  53. 53. FIGURE 55: Simulink Model of WMR Results: The look ahead point should trace an ellipse of given parameters. The motion of the WMR is simulated and the generalized coordinates and velocities are plotted as shown below: FIGURE 56: X and Y Coordinates of the Look ahead point 53
  54. 54. FIGURE 57: and vs time FIGURE 58: and vs time 54
  55. 55. FIGURE 59: ∅ vs time 55
  56. 56. Symbolic Modeling using MapleSim MapleSim is a complete environment for modeling and simulating complex multidomain physical systems. It allows building component diagrams that represent physical systems in a graphical form. Using both symbolic and numeric approaches, MapleSim automatically generates model equations from a component diagram and runs high-fidelity simulations. We can generate the model equations and then use them for post processing in the Maple environment without having to worry about deriving them. Implementation  Create the Model in MapleSim  Obtain the multibody equations using the multibody equations template  Copy the equations in a new document folder  Solve the ODE‟s  Post process Case Study 1: Simple Pendulum Mass of the Link m = 2Kg Length of the Link l = 2m TABLE 8 Creating MapleSim Model 56
  57. 57. FIGURE 60: MapleSim Model of Simple Pendulum Obtaining the multibody equations and solving them The multibody equations of the above model are obtained from the multibody equations template and they are copied into a new Maple document and solved. Below attached in the Maple document showing the same: Mass Matrix (M) Equations of Motion (EOM) 57
  58. 58. Solution of EOM Results FIGURE 61: vs time 58
  59. 59. FIGURE 62: Animation of the Pendulum Initial Condition: = pi/4 from vertical 59
  60. 60. Results Simulating the system for 10 secs in Matlab and in MapleSim gives us the following results: FIGURE 63: X-Y position of the end effector in Matlab and in MapleSim FIGURE 64: in Matlab and in MapleSim 60
  61. 61. FIGURE 65: in Matlab and in MapleSim 61
  62. 62. Case Study 2: Two Link Mechanism To simulate the motion of a double pendulum under gravity in Matlab and MapleSIm and compare the results. Link Lengths L1 = 2, L2 = 1 Distance of Mass centers Lci = Li/2, i = 1,2 Link Masses and Inertias M1 = 2, M2 =1 , Ji = Mili^2/3 TABLE 9 Creating MapleSim Model FIGURE 66: MapleSim Model of Double Pendulum Obtaining multibody equations and solving them Mass Matrix 62
  63. 63. 63
  64. 64. Equations of Motion 64
  65. 65. Results 65
  66. 66. FIGURE 67: vs time 66
  67. 67. FIGURE 68: vs time 67
  68. 68. FIGURE 69: Animation of Two link mechanism Initial conditions: 1 = -pi/4; 2 = -pi/4; Simulation Time = 10 secs. 68
  69. 69. FIGURE 70: X-Y Position of the End-Effector in Matlab and MapleSim FIGURE 71: vs time 69
  70. 70. FIGURE 72: vs time FIGURE 73: vs time 70
  71. 71. FIGURE 74: vs time 71
  72. 72. Case Study 3: Four-Bar Mechanism Dynamic Analysis FIGURE 68: A Fourbar Mechanism The dimensions and mass properties of the Four Bar mechanism are given in the table below: Link Lengths 0 = 4, 1 = 1, 2 = 5, 3 = 4 Distance of mass centers = , = 1,2,3 2 Link masses = 1, = 1,2,3 Moment of inertias m i l2 i = , i = 1,2,3 12 Gravitational acceleration = 9.81 −2 Torque 6 N.m Lagrangian Formulation 72
  73. 73. The Equations of Motion(EOM) for the overall system are derived by treating the four-bar linkage as being composed of two dyads, (a) Dyad 1 is the 2 DOF left chain ( combination of link 1 and 2 and b) Dyad 2 is the 1 DOF right chain (only link 3 ), as shown in the figure below. FIGURE 69: A Fourbar Mechanism being decomposed into two Dyads The EOM of each (unconstraint) subsystems are derived independently by the Lagragian‟s method, where For Dyad 1 73
  74. 74. For Dyad 2 The constraint equations are then obtained from the requirement that Dyad 1 and Dyad 2 to stay connected at the cut joint, which can be expressed in matrix form as We can obtain the Jacobian matrix of the constraint as The EOM of the combined overall system can be finally be modularly constructed as an index-3 DAEs as 74
  75. 75. Where From the constraint equation we can get We can also determine as That is It means that the above expression is always zero. Since we choose θ1 as the independent variable, we can rewrite And we can also get Where S is a 3×1 matrix, 75
  76. 76. EOM can be rewritten to We can show that, We can also derive that We multiply ST to both sides of EOM , In forward dynamics, we take the actuate torques as input, and take the motion of the joint as output. We can find the motion according to the equation In inverse dynamics, we take the joint motion as input, and take the actuate torques as output, Analytical Model Our goal is to be able to implement the model described before in a real-time application. We use Matlab for our Simulation. We use the Compliance based method to formulate the EOM into state space form and simulate for 10secs. And plot the angular orientations and end effector trajectory. 76
  77. 77. MapleSim Model The model looks as shown in the figure below FIGURE 70: MapleSim Model of a Fourbar Symbolic multibody equations Mass Matrix (M) 77
  78. 78. Equations of Motion (EOM) 78
  79. 79. 79
  80. 80. Results FIGURE 71: X-Y trajectory of the point of interest in Matlab and MapleSim FIGURE 72: vs time 80
  81. 81. FIGURE 73: vs time FIGURE 74: vs time 81
  82. 82. Conclusion Thus, we can clearly see that the analytical formulation of the above examples is a very tedious and time consuming process and virtual prototyping techniques improve the ease of modeling with reasonable errors. MapleSIm also provides symbolic modeling techniques wherein the dynamic equations of motion of a given system can be extracted out which can be a novel area of interest. 82
  83. 83. References  “Control of Mechanical Systems with Rolling Contraints: Application to Dynamic Control of Mobile Robots”, Nilanjan Sarkar,Xiaoping Yun and Vijay Kumar  83

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