inertia.md

Tutorial: Configuring Inertia for the Virtual Shake Robot (VSR)

This tutorial will guide you through calculating the inertia for a rectangular box, configuring these values in a YAML file for the Virtual Shake Robot (VSR), and understanding how PyBullet reads these values in the createMultiBody function.

1. Calculating Inertia

Inertia for a Rectangular Box

The moments of inertia for a rectangular box aligned with the coordinate axes can be calculated using the following formulas:

$$ I_{xx} = \frac{1}{12} m (h^2 + d^2) $$

$$ I_{yy} = \frac{1}{12} m (w^2 + d^2) $$

$$ I_{zz} = \frac{1}{12} m (w^2 + h^2) $$

Where:

• ( m ) is the mass of the box.
• ( w ) is the width (x-dimension).
• ( h ) is the height (y-dimension).
• ( d ) is the depth (z-dimension).

Example Calculation

Given a rectangular box with:

• Dimensions: [10.0, 10.0, 0.1] (width, height, depth)
• Mass: 10000.0

The moments of inertia are calculated as follows:

For the x-axis (( I_{xx} )) and y-axis (( I_{yy} )):

$$ I_{xx} = I_{yy} = \frac{1}{12} \times 10000.0 \times (10.0^2 + 0.1^2) = 8334.167 $$

For the z-axis (( I_{zz} )):

$$ I_{zz} = \frac{1}{12} \times 10000.0 \times (10.0^2 + 10.0^2) = 166666.67 $$

Since it is a simple geometric box and symmetrically distributed around its center of mass, the other non-diagonal elements are zero.

2. Configuring Inertia in VSR

The calculated moments of inertia need to be configured in the YAML file used by the VSR. Here is how you can specify the configuration:

YAML Configuration

Create or modify the YAML configuration file as follows:

simulation_node:
  ros__parameters:
    structure:
      world_box:
        dimensions: [50.0, 14.0, 2.0]
        mass: 0.00
        inertia: [0.0, 0.0, 0.0]

      pedestal:
        dimensions: [10.0, 10.0, 0.1]
        mass: 10000.0
        inertia: [8334.167, 8334.167, 166666.67]

3. Setting the Correct Parameters in the setJointMotorControl2

Calculation of Maximum Force and Maximum Velocity

Objective

To determine the maximum force and velocity required for controlling the pedestal in the simulation based on given amplitude (A) and frequency (F) values.

Parameters

• Mass of the pedestal (m): 10000 kg
• Amplitude (A): Variable
• Frequency (F): Variable

Formulae Used

1. Acceleration Function: $$ a(t) = 4\pi^2 F^2 A \cos(2\pi F t) $$

   • (a(t)): Acceleration as a function of time.
   • (A): Amplitude.
   • (F): Frequency.
   • (t): Time.

2. Newton's Second Law: $$ F = ma $$

   • (F): Force.
   • (m): Mass.
   • (a): Acceleration.

3. Velocity Function: $$ v(t) = 2\pi AF \sin(2\pi F t) $$

   • (v(t)): Velocity as a function of time.

Calculations

Maximum Acceleration and Force

The maximum acceleration occurs when

$$ \cos(2\pi F t) = 1 $$ $$ a_{max} = 4\pi^2 F^2 A $$

For a given mass ((m)) of 10000 kg, the maximum force can be calculated using Newton's Second Law:

$$ F_{max} = m \times a_{max} $$

Example Calculation for ((A = 1, F = 16)):

1. Calculate maximum acceleration: $$ a_{max} = 4\pi^2 \times 16^2 \times 1 = 10093.5104 , \text{m/s}^2 $$

2. Calculate maximum force: $$ F_{max} = 10000 \times 10093.5104 = 100935104 , \text{N} $$

Maximum Velocity

The maximum velocity occurs when (\sin(2\pi F t) = 1):

$$ v_{max} = 2\pi AF $$

Example Calculation for ((A = 5, F = 5)):

1. Calculate maximum velocity: $$ v_{max} = 2\pi \times 5 \times 5 = 157.08 , \text{m/s} $$

Summary of Calculated Values

Amplitude (A)	Frequency (F)	Maximum Acceleration (m/s²)	Maximum Force (N)	Maximum Velocity (m/s)
1	16	10093.5104	100935104	100.53
10	1	3959.20139	39592013.9	62.83
2	2	3159.36111	31593611.1	25.13
2	1	789.18028	7891802.8	12.57
5	5	19739.5064	197395064	157.08

Decision on Maximum Force

The maximum force calculated for ((A = 1, F = 16)) is (100935104) N, which is the highest among the test cases. Therefore, we decided to use this value to ensure the controller can handle the most extreme scenario.

Implementation in Code

The calculated maximum force is used in the setJointMotorControl2 function to ensure the joint motor can reach the desired positions and velocities:

p.setJointMotorControl2(
    bodyUniqueId=robot_id,
    jointIndex=0,
    controlMode=p.POSITION_CONTROL,
    targetPosition=positions[i],
    targetVelocity=velocities[i],
    force= 5 * 10**8,  # Maximum force set based on calculations
    maxVelocity=200,  # Set maximum velocity
    physicsClientId=client_id
)

Tutorial: Calculating Inertia for a Mesh File (.obj) in Autodesk Fusion 360

Step 1: Import the Mesh File

1. Open Fusion 360: Launch Autodesk Fusion 360 on your computer.
2. Create a New Project: If you don’t already have a project, create a new one by clicking on the New Project button in the Data Panel.
3. Import the Mesh:
   • Go to the Insert menu.
   • Select Insert Mesh.
   • Browse to your .obj file and select it.
   • Click Open to import the mesh into your project.

Step 2: Convert the Mesh to a Solid

1. Prepare the Mesh:

   • Right-click on the mesh in the Browser pane.
   • Select Mesh to BRep (Boundary Representation).
   • Fusion 360 will prompt you to convert the mesh. Confirm and proceed.

2. Edit the Mesh (if necessary):

   • Use the Repair tool to fix any issues with the mesh. This is available under the Mesh tab.
   • Simplify the mesh if it’s too detailed. This can be done using the Reduce tool.

Step 3: Calculate the Inertia

1. Create a New Component:

   • Right-click on the root component in the Browser pane.
   • Select New Component.

2. Move the Solid into the New Component:

   • Drag the solid body from the Browser pane into the newly created component.

3. Go to the Inspect Menu:

   • Click on the Inspect menu.
   • Select Physical Properties.

4. Select the Solid:

   • Click on the solid body you want to analyze.

5. Calculate Properties:

   • Fusion 360 will display the mass, volume, center of mass, and moments of inertia for the selected solid.
   • You can view the moments of inertia in the results window.

6. Export the Inertia Data (Optional):

   • Click on Export to save the physical properties data to a CSV file for further analysis or record-keeping.

Step 4: Validate and Use the Inertia Data

1. Check Units:

   • Ensure the units of measurement are consistent with your requirements (e.g., kg·m² for moments of inertia).
   • You can change units in the Document Settings at the bottom of the Browser pane.

2. Use the Data:

   • The moments of inertia provided can be used for further simulations, control algorithms, or integrating with other software like PyBullet.

Tips and Tricks

• Simplification: Before converting the mesh to a solid, it might be helpful to simplify the mesh to reduce complexity and improve performance.
• Mesh Repair: Always check for and repair any errors in the mesh to ensure accurate physical properties.
• Fusion 360 Help: Utilize Fusion 360’s help resources and community forums if you encounter any issues.

This tutorial should guide you through the process of calculating inertia for a mesh file in Autodesk Fusion 360. If you need more advanced features, Autodesk’s official documentation and support can provide additional assistance.