Bodies and joints

Bodies and joints define the tree structure of the multibody systems

Bodies

  • They are fully defined by:
    • The mass
    • The position of center of mass
    • The inertia matrix
  • Particular points of a body can be identified using anchor points
  • Each body must be connected to one and only one joint

Joints

  • A joint defines the relative motion between two bodies
  • A joint can be attached to
    • The base
    • An anchor point
    • An other joint

Example: the pendulum-spring-mass system

The initial example model consists of a pendulum with a mass sliding along

Pendulum spring illustration

REMARK:

In Robotran right handed coordinates convention is used. In this case, on the figure, positive rotations are anticlockwise

Model data

The example model is composed of the following elements:

  • Two bodies
    • the pendulum
      • Mass: 5 kg
      • Inertia along axis \(\hat{I}_{2}\): 0.1 kg.m2
    • the slider
      • Mass: 2 kg
  • Two joints
    • A revolute joint between the pendulum and the base
    • A prismatic joint between the pendulum and the slider
  • A linear spring-damper element
    • Will be implemented as a joint force
      • Stiffness: 100 N/m
      • Damping: 2 Ns/m
      • Free length: 0.1 m

Geometrical data is given in the figure above.

The following initial conditions are considered:

  • Rotation of the pendulum: 1 rad
  • Translation of the slider with respect to point A: 0.2 m

Step 1: Draw your multibody system

Draw a base body:

  • Click the body button
    • Select a shape
  • Click in the drawing area (for a rectangle: click twice: once for 1st corner and once for 2nd corner)
  • In the right panel, set the z component of the gravity to 9.81m/s2
Create a new body by clicking the Body menu

Body drop down menu

REMARK:

The shape of bodies in the 2D diagram does not influence the dynamics

Draw a R2 joint:

  • Click the joint button
  • Select R2
  • Click in the drawing area to draw the joint
  • Activate the joint properties panel by clicking on the joint
  • Give a name to the joint, for instance “R2_pendulum”
Create a new joint by clicking the Joint menu

Joint drop down menu

REMARK:

In the R2 naming:

  • “R” stands for revolute
  • “2” stands for axis 2 (y-axis)

WARNING:

be careful when giving name to the elements: use only alphanumeric characters, always start with a letter (as for any variable in a program code).

Draw the pendulum body:

  • Same procedure as for the base body
  • Attach this body to the R2 jointSelect a shape
  • Activate the body properties panel by clicking on the body
  • In the right panel, give a name to the body and fill the dynamic properties
    • Center of Mass coordinates in the body-fixed frame, it is aligned with the local Z axis
    • Inertia matrix in the body-fixed frame, with respect to the center of mass
Create the pendulum body

Pendulum body snapshot

REMARK:

When drawing a body, you can select the joint it will be attached to by coming close to this joint (which will be highlighted).

REMARK:

For each body, there is a body-fixed frame: this frame move with the body and is fixed at the position of the parent joint. Its axis are aligned with the axis of the parent body when joint positions between the 2 bodies are set to 0.

Draw an anchor point on the pendulum body:

  • Click the Anchor Point button
  • Click in the drawing area on the pendulum body
  • Fill the point coordinates in the right panel (coordinates in the body fixed frame, it is aligned with the local Z axis)
  • This anchor will be reference point for the prismatic joint.
Create an anchor point

Anchor point snapshot

Complete the diagram:

  • Draw a T3 joint attached to the anchor point using the same procedure as for the R2 joint
  • Draw a body for the slider (Don’t forget to fill in its dynamical properties)
Full MBsysPad diagram of the Pendulum sping example

Snapshot of the complete diagram

REMARK:

In the T3 naming:

  • “T” stands for translation
  • “3” stands for axis 3 (z-axis) of the pendulum fixed-frame

Set initial conditions for the joints:

  • Click on the joint you want to edit
  • Enter the initial value at the bottom of the right panel

Save your project

REMARK:

All the data entered in MBsysPad are saved in a *.mbs file which is located in the dataR subfolder of your project. The *.mbs file uses an XML format.

Step 2: Generate your multibody equations

  • Click on the menu Tools > Generate Symbolic Files
Access the symbolic generation interface via the menu Tools/Generate Symbolic Files

Snapshot of the Generate symbolic file menu

  • Enter your username and password
  • Select your programming language: Python
    • The language is case-sensitive;
    • Several language can be generated, split them with a coma;
    • The language of the template has te be included;
  • Keep other options to default
  • Click on the Generate button
  • Check that symbolic files have been downloaded to the symbolicR subfolder of your project
Enter your username and password in to the symbolic generation gui

Snapshot of the symbolic generation gui

REMARK:

If you don’t have an account you can either use demo account (usr = demo, pw = demo) or ask one at info@robotran.be

Step 3: Write your user function

Python section:

  • Implement the spring-damper law
    • Edit the user_JointForces function (located in the joint_forces.py file from the userfctR subfolder of your project)
    • Write the force equations with a stifness K of 100 [N/m] and a damping C of 2 [Ns/m]
def user_JointForces(mbs,tsim):
  #...
  # set the joint force in joint 2
  id = 2
  K = 100
  C = 2
  L0 = 0.1
  mbs.Qq[id] = - ( K*(mbs.q[id]-L0) + C*mbs.qd[id] )
  #...

NB :

  • mbs.q is the joint position
  • mbs.qd is the joint speed

REMARK:

A joint forces is a force acting along the axis of a joint. It is thus:

  • a force for a prismatic joint,
  • a torque for revolute joint.

Joint forces corresponds to the Qq vector in the equation of motion:

\(\begin{matrix} M(q)\ddot{q} + c(q, \dot{q}, frc, trq, g) = Q(q, \dot{q}) + J^T\lambda \\ h(q) = 0 \end{matrix}\)

Qq is the force/torque acting from the parent body to the child body. The reaction is automatically taken into account by the multibody formalism.

Step 4: Run your simulation

Open the main.py file located in your workR folder:

  • Set the end of simulation at 5 seconds : dirdyn.setTspan(0.0,5.0)
  • Check that the name of the project at the line mbs=Data(...) is the same as your .mbs file
    • MBsyspad should have create the file with your project name already included.
    • The name has to include the .mbs extension: mbs=Data(‘example.mbs’)
    • You have to update the name if you retrieve the file from an older project.
  • Run the file : python main.py

Step 5: Animate your 3D MBS

Draw your MBS system in 3D

You can customize the 3D view of your system as follows:

  • Open your *.mbs file in MBsysPad
  • Click on the Design 3D model button
Open the Design 3D frame using the toolbar button

Snapshot of the Design 3D icon

  • Activate the body you want to edit
    • Click on the body in the 2D view
    • Or go to the 3D view and, in the right panel, use the drop down menu of the Comp tab
Select the object you want to edit

Snapshot of the 3D object selection menu

  • Add a shape and edit its properties
    • The shape inline allow to add a 3D part exported from CAD software. The part must be saved in the VRML format (extension *.wrl). Only VRML 2.0 or VRML 97 are supported (not VRML 1.0).
Edit the 3D properties of a body and add shapes to its representation

Snapshot of the 3D properties panel for bodies

  • Use the ViewPt to add registered viewpoint
  • Use the Light tab to add lights to the scene

REMARK:

Attaching a viewpoint or a light to a joint will make them follow the motion when animating the model.

Animate your results

You can view a 3D animation of the system using MBSysPad as follows:

  • Open your *.mbs file in MBsysPad
  • Click on the Animate 3D model button
Open the Animate 3D frame using the toolbar button

Snapshot of the Animate 3D icon

  • Load the result file
    • Click on the Open button
    • Select the *.anim file located in the animationR subfolder of your project
Select the animation file via the open file button

Snapshot of the Open file button

  • Use the control buttons to run the animation
Control the animation using the toolbar

Snapshot of the Control toolbar of the Animate 3D frame

REMARK:

Navigating in the 3D view:

  • Rotation : Mouse motion + left button
  • Zoom : Mouse motion + middle button (OR ALT + mouse motion + single button)
  • Translation : Mouse motion + right button

Check the results

Plot the graph of the joint position (results ares avilaible in resultsR/ folder) and check your results with the following graph.

Plot the time history of the joint position of the pendulum spring example

Time history of the joint position