Mapping 3D Spatial Velocity Between Coordinate Frames
In the previous lecture, we looked at transforming spatial velocity from one coordinate frame to another. We did this for the two-dimensional case and we learned that the relationship between velocities is given by a Jacobian matrix which is actually a function of the relative orientation between the two coordinate frames.
For three dimensions, things are actually quite similar. So, here is a three-dimensional reference coordinate frame, a point which has a spatial velocity with respect to Frame A. And, here is Frame B and we can describe the relative pose between Frame A and Frame B. We can show in red the spatial velocity with respect to Frame B and the relationship between the two is given by a Jacobian matrix. For the three-dimensional case, the Jacobian matrix is 6 x 6. And, once again, it is a function only of the relative orientation between the two coordinate frames. It’s a function of rotation matrix from Frame B to Frame A.
Here is an example of an industrial robot. And, if I apply a particular set of robot joint angle velocities, I obtain a particular spatial velocity for the robot end effector and those two quantities are related by the manipulator, Jacobian.
Now, this assumes that the robot end effector velocity is defined with respect to the world coordinate frame, denoted zero here. It’s often very useful to describe the velocity of the end effector with respect to the end effector’s coordinate frame. I have denoted here as Frame 6.
Now, we can write this in terms of a different Jacobian matrix. This Jacobian matrix relates the robot joint angle velocity to the robot end effector velocity expressed in terms of Frame 6.
We can write an expression for the end effector’s spatial velocity with respect to Frame 6 in terms of two Jacobian matrices. These Jacobian is the standard robot manipulated Jacobian matrix that we have talked about previously. And, this Jacobian matrix is responsible for transforming velocity from Frame 0 to Frame 6 and it is a function of the relative pose from Frame 6 to Frame 0. And, that Jacobian matrix, as we introduced in the previous slide, is a function simply of the rotational paths of that relative pose.
We previously learnt how to derive a Jacobian which relates the velocity of a point, defined relative to one coordinate frame, to the velocity relative to a different coordinate frame. Now we extend that to the 3D case.
This content assumes high school level mathematics and requires an understanding of undergraduate-level mathematics; for example, linear algebra - matrices, vectors, complex numbers, vector calculus and MATLAB programming.
3D mapping, another wonder of modern technology, is a new machine vision technology in the field of cartography. While we are already familiar with 2D maps and 3D vision, it is not hard to imagine how a 3D map looks like? Don’t let the curvature of Earth hinder your view and zoom in close to your location using 3D mapping.
3D mapping means profiling of objects in three dimensions to map the objects in real-world. There are several ways for a 3D profiling of an area or object, such as using a pair of a stereo camera. Another method is to measure the depth of an object or feature from focus. You can also create a 3D effect by detecting the light field by plenoptic methods or using active 3D imaging techniques.