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Project: Graphically Resize Inertia

Gazebo Design Document

Overview

When modeling a robot, one must choose inertial parameter values for each rigid body (or Link) in the system. These parameters include the mass, center of mass location, and the six components of the symmetric 3x3 moment of inertia matrix. While intuition can often be used to estimate the mass and location of the center of mass, it is very difficult to estimate inertia matrix parameters, which depend on both the mass and size of an object.

For this reason, visualization of the moment of inertia was added to Gazebo in pull request 745 (see also issue 203). While it is useful to visualize the inertia, it would be even more useful to be able to modify the inertia using interactive markers. Gazebo currently has the ability to resize simple shapes; this should be extended to modify the inertia as well.

Requirements

  1. When resizing a simple shape in simulation mode, scale the inertia values accordingly. Note that this was implemented for resizing in the Model Editor in gazebo pull request 1836.

  2. When modifying the mass or density of an object in the Model Editor, there should be an option for scaling the moment of inertia according to the new values. Perhaps the moment of inertia could be "linked" to the mass / density parameters?

  3. Allow the moment of inertia values to be modified in the Model Editor by attaching interactive markers to the inertia visualization and scaled in a similar manner to the other resize tool.

  4. Properly account for off-diagonal inertia terms. This requires the ability to diagonalize a 3x3 symmetric positive definite matrix. This is discussed in issue 880 and has been implemented in ign-math pull request 82.

Box of equivalent inertia

To visualize inertia, we utilize the concept of a box of equivalent inertia. Given the mass m and principal moments of inertia Ixx, Iyy, and Izz, the dimensions of a box of uniform density are computed that has an equivalent moment of inertia. For example, a box with dimensions dx, dy, and dz has the following moment of inertia components:

Ixx = m/12 (dy^2 + dz^2)
Iyy = m/12 (dz^2 + dx^2)
Izz = m/12 (dx^2 + dy^2)

The equations can be inverted to express the box dimensions as a function of mass and inertia components:

dx = sqrt(6/m (Izz + Iyy - Ixx))
dy = sqrt(6/m (Ixx + Izz - Iyy))
dz = sqrt(6/m (Iyy + Ixx - Izz))

These calculations are currently used to visualize the inertia of an object as a pink box.

inertia box of a sphere

Inertia values for other shapes can be found on wikipedia. These could be used to derive alternative visualizations. The ellipsoid, for example, is sometimes used for inertia visualization.

Architecture

Modifying inertia properties graphically will consist of GUI elements that communicate changes in inertial parameters to the gazebo server over the ~/model/modify topic. This is a proven interface, it simply needs tests to be added for inertia parameter modification.

Additionally, the mathematical operators in the gazebo::physics::Inertial class will be moved to the Ignition Math library. This class stores and allows computation about the mass, moment of inertia, and center of mass location. To address issue 880, an additional subclass MassMatrix3 will be defined that contains the scalar mass and symmetric moment of inertia matrix components. It will provide methods for diagonalizing the matrix to determine the principal moments of inertia and identify the orientation of the principal axes, based on this paper.

class MassMarix3<T>
{
  // Scalar mass
  private: T mass;
  // Diagonal components of moment of inertia matrix
  private: Vector3<T> Ixxyyzz;
  // Off-diagonal components of moment of inertia matrix
  private: Vector3<T> Ixyxzyz;
};

class Inertial<T>
{
  // Pose offset from link frame to inertial frame.
  // The inertial frame position specifies the center of mass location
  // and the orientation is used to interpret the inertia matrix.
  private: Pose3<T> pose;
  // Mass and moment of inertia matrix expressed in inertial frame.
  private: MassMatrix3<T> massMatrix;
};

Interfaces

The graphical resizing of inertia can be implemented using the graphical markers used for the simple shape resize tool, perhaps with pink arrow heads:

resizing a sphere

The MassMatrix3 class will provide functions for converting between the mass and moment of inertia matrix and the size of an equivalent box of uniform density. This will allow the GUI tools to perform these calculations without re-implementing the mathematical solution.

Performance Considerations

The computational performance of the matrix diagonalization method can be evaluated, but it is not expected to be used frequently in a computational loop, so it should not affect performance. If the inertia matrix has no off-diagonal terms, then it is already diagonalized and the computation can be skipped, so this should be checked by the algorithm.

Tests

  1. Change inertial parameters with ~/model/modify for each physics engine.
    1. Test inertial accessors: in world with no gravity, modify mass, inertia components, and center-of-mass location and verify that C++ accessors confirm the changes.
    2. Test mass changes: in world with gravity, create a balanced seesaw with stacked boxes then increase the mass on one side and verify that it tips over.
    3. Test center-of-mass changes: in world with gravity, create a balanced seesaw with stacked boxes then move the center of mass of the seesaw plank towards one of the balanced boxes. Verify that it tips over.
    4. Test moment of inertia changes: in world with gravity, create swinging pendulum and measure its oscillation frequency. Increase the moment of inertia and verify that its oscillation frequency decreases.
  2. Unit tests for MassMatrix3 and Inertial classes
    1. Test Accessors and Modifiers.
    2. Test matrix diagonalization with diagonal matrices.
    3. Test matrix diagonalization with non-diagonal matrices.
    4. Test conversion functions for equivalent box of uniform density.
  3. GUI tests in Simulation:
    1. Check that the visuals are updated after a ~/model/modify message is published.
    2. Check that the left panel is updated after a ~/model/modify message is published.
  4. Model Editor tests:
    1. Check that values in the link inspector are updated when the user resizes the inertia in the 3D view.
    2. Check that inertia visualization (not yet present in the editor) updates correctly when the inertia is changed in the link inspector.

Pull Requests