MASTERCLASS

# Velocity kinematics in 3D

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In the last lecture, we talked about the relationship between the velocity of a robot's joints and the velocity of the robot end-effector, the rate of change of the pose of the robot end-effector. And, we talked previously about these two robotic problems, the forward kinematic problem and the inverse kinematic problem and then we introduced this velocity transform problem.

In our last lecture, we did everything in two-dimensional space. This we have seen a number of times through this course. Things are pretty easy in two dimensions. They get much more complex in three dimensions. In this lecture, we are going to do it in three dimensions.

So now, we are going to define what does it mean, this psi dot, in terms of three-dimensional pose changing with time. So, first, we’re going to describe what mathematical object this psi dot is and then we’re going to work on the relationship between joint angle rates and pose rate.

Now, as we do this, there are some advanced concepts and, if you are not particularly strong at maths, you don’t have background in differential calculus, your not too comfortable with linear algebra and we've done a bit of linear algebra already. We used matrices and vectors, but this is going to be perhaps matrix and vector used at a more advanced level.

If you are not too comfortable with these concepts, you might struggle with this particular lecture. So, your options are just to try and skim through the lecture, observe the gist of what I’m trying to say even if you don’t get all of the details. Maybe look at some, some additional textbooks. Do some more reading to get the, the prerequisites that you are going need to understand that. So, just a little bit of a, of a warning before we get in to the details of it.

We will learn about the relationship, in 3D, between the velocity of the joints and the velocity of the end-effector — the velocity kinematics. This relationship is described by a Jacobian matrix which also provides information about how easily the end-effector can move in different Cartesian directions. To do this in 3D we need to learn about rate of change of orientation and the concept of angular velocity.

### Skill level

Undergraduate mathematics

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.