# 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 parent joint

## Joints¶

• A joint defines the relative motion between two bodies

• A joint can be attached to

• The base

• An anchor point

• Another joint

## Example: the pendulum-spring-mass system¶

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

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”

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 joint

• 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

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.

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)

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

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

Snapshot of the Generate symbolic file menu

• 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

Snapshot of the symbolic generation gui

REMARKS:

If you don’t have an account you can either use demo account (usr = demo, pw = demo) or ask one via the online form Need an account ? on the Robotran website

You can save you configuration (username, password, language …) by clicking on the “Save To File” button

You can reset your parameters by clicking on “Settings->Reset password for symbolics”

### Step 3: Write your user function¶

• 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_data,tsim):
#...
# set the joint force in joint 2
id_j = 2
K = 100
C = 2
L0 = 0.1
mbs_data.Qq[id_j] = - ( K*(mbs_data.q[id_j]-L0) + C*mbs_data.qd[id_j] )
#...


NB :

• mbs_data.q is the joint position

• mbs_data.qd is the joint speed

REMARK:

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

• a force for a prismatic joint,

• a torque for revolute joint.

Joint forces correspond to the Qq vector in the equations 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 : mbs_dirdyn.set_options(tf=5.0)

• Check that the name of the project at the line mbs_data = Robotran.MbsData(...) is the same as your .mbs file

• 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 : python3 main.py

• If you do not manage to run your code, please check KnownTroubleshootings.

### Step 5: Animate your 3D MBS¶

#### Draw your MBS system in 3D¶

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

• Click on the Design 3D model 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

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).

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.

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

• Click on the Animate 3D model button

Snapshot of the Animate 3D icon

• Click on the Open button

• Select the *.anim file located in the animationR subfolder of your project

Snapshot of the Open file button

• Use the control buttons to run the animation

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.

Time history of the joint position