Demystifying the Jacobian Derivative in Drake

Описание: Discover how to compute the `Jacobian Derivative` in Drake effectively. This guide provides clarity and structured steps for feedback linearization.
---
This video is based on the question https://stackoverflow.com/q/65675339/ asked by the user 'Murt' ( https://stackoverflow.com/u/9717983/ ) and on the answer https://stackoverflow.com/a/65675828/ provided by the user 'Mitiguy' ( https://stackoverflow.com/u/13470670/ ) at 'Stack Overflow' website. Thanks to these great users and Stackexchange community for their contributions.

Visit these links for original content and any more details, such as alternate solutions, latest updates/developments on topic, comments, revision history etc. For example, the original title of the Question was: Derivative of Jacobian in drake

Also, Content (except music) licensed under CC BY-SA https://meta.stackexchange.com/help/l...
The original Question post is licensed under the 'CC BY-SA 4.0' ( https://creativecommons.org/licenses/... ) license, and the original Answer post is licensed under the 'CC BY-SA 4.0' ( https://creativecommons.org/licenses/... ) license.

If anything seems off to you, please feel free to write me at vlogize [AT] gmail [DOT] com.
---
Understanding the Challenge: Jacobian Derivative in Drake

When working with robotics frameworks like Drake, developing control systems often requires a robust understanding of dynamics, particularly when it comes to calculating various terms integral to motion control. One of the key components in this process is the Jacobian and its derivative.

In this post, we'll address:

How to compute the mass matrix and gravity terms using the Drake framework.

The methods to calculate Coriolis terms effectively.

A detailed breakdown of how to derive the Jacobian and its time-derivative.

Extracting Mass and Gravity Terms

Before diving into the Jacobian, it's crucial to understand how to calculate the mass matrix and gravity terms. Here’s how you can do it:

1. Mass Matrix Calculation

To compute the mass matrix, you can use:

[[See Video to Reveal this Text or Code Snippet]]

This command retrieves the mass distribution of your multibody system.

2. Gravity Terms

For calculating gravitational effects, utilize:

[[See Video to Reveal this Text or Code Snippet]]

This will return the generalized forces acting on the system due to gravity.

Addressing Coriolis Terms

While the commands above cover mass and gravity, the challenge remains in calculating the Coriolis terms.

Alternative Approach for Coriolis and Gravity

A straightforward method to compute gravity and Coriolis terms jointly involves:

[[See Video to Reveal this Text or Code Snippet]]

This method will allow you to consider both effects without ambiguity in your computations.

Calculating the Jacobian

Now that we've addressed mass, gravity, and Coriolis terms, let's explore how to calculate the Jacobian. We must establish the relationship between the end-effector's position and the generalized coordinates of the system.

Step-by-Step Jacobian Calculation

Identify the End-Effector Link
Access the end-effector by its name:

[[See Video to Reveal this Text or Code Snippet]]

Defining Frames
Establish the end-effector’s frame and the world frame:

[[See Video to Reveal this Text or Code Snippet]]

Set Up Position
Define the position vector from the end-effector frame to the origin:

[[See Video to Reveal this Text or Code Snippet]]

Calculate Jacobian
Finally, you can calculate the Jacobian matrix:

[[See Video to Reveal this Text or Code Snippet]]

The Jacobian Derivative: The Next Step

The final piece of the puzzle is obtaining the Jacobian derivative (J̇).

What Do You Really Need?

Before you start looking for J̇, consider whether you need the time-derivative of the Jacobian itself or its product with the velocities, denoted as J̇*v.

Options for Calculating the Jacobian Derivative:

Use MultibodyPlant::CalcBiasSpatialAcceleration() if you need spatial acceleration.

For translational parts specifically, you can use MultibodyPlant::CalcBiasTranslationalAcceleration().

Implementation Example:

Here’s how to use the CalcBiasSpatialAcceleration method:

[[See Video to Reveal this Text or Code Snippet]]

Conclusion

In summary, obtaining the Jacobian Derivative in Drake can be straightforward once you understand the steps involved. Compute mass and gravitational effects, work with the Jacobian for translational velocity, and then leverage the necessary methods for derivatives effectively.

If you have any questions or need further clarification, feel free to drop a comment!