IntroductionToRobotics-Lecture04

Instructor (Oussama Khatib):All right. Let’s get started. So today video segment is about a small device called the Hummingbird. The Hummingbird was developed at IBM Watson Research Center, and it was published in the proceeds of ICRA 1992. Female Speaker:

Instructor (Oussama Khatib):Unbelievable. Female Speaker:

The Hummingbird mini-positioner resulted from the interdisciplinary team effort of these and many other contributors.

Student:The distance?

Instructor (Oussama Khatib):Distance? Basically, these axes are maintaining a distance, right? And in general, these axes are not parallel, so there is a tilt between them, and that tilt is going to be maintained. And there are some offsets that will be introduced. There is an angle that is taking place that we cannot see it with just the axis. We need to assign the frame, and we will start to see that relationship. So the link description, I’m going to take two axes, and arbitrary axes so that we will not just take the parallel axis case. So axis I minus one and axis I are connected somehow through this link. So if we take a link, at the extremities of the link we have the joint axes. So what are the things that are constant? So you said distance? How is that going to define the distance between two axes? Come on. Faster.

Student:Perpendicular.

Instructor (Oussama Khatib):So perpendicular. Perpendicular to a plane, perpendicular to an axis, but we have two axes, so it is a common perpendicular, right? Something that is perpendicular to both that would measure that distance.

So if we take this common perpendicular to both axes, then we have a sort of – this is going to be unique, right? Except if the axes are parallel. Then you have infinite way of placing that common perpendicular. Okay, you agree with that selection? Does it make sense? We take the common perpendicular, and that will give us this distance. So we call it A – so now you have to pay attention to the notation because we’re going to describe link I minus one with the parameter A I minus one, which is the common perpendicular to those two axes. So I minus one is the common perpendicular between axes I minus one and I. All right. What else do we need to introduce? So if I slide axis I along this common perpendicular, and I come to the intersection, there will be an angle, a twist angle. This angle, you see it? We slide it up to the intersection, and there will be an angle. We call it the link twist, and it is the parameter alpha I minus one, which measures this angle. And what we will do is we measure the angle along the vector A I minus one in the right hand sense. And you’re going to learn how to use your – everyone knows how to measure the angle in the right hand sense? Just make sure you use your right hand. It happens. Okay. So we have two parameters. In fact, we’re going to see that in total we need four parameters. One of them is variable, the joint angle or the joint displacement if it’s a prismatic joint. And now we identify two. Alpha and A are constant all the time, so once you design your robot, these alphas, Alpha 0, Alpha 1, are going to be constant. The same for A. Now if we look at most mechanisms, we’re going to see that the axes are not always apart. Most of the time they are parallel, and sometimes they are intersecting. So if we take the PUMA, we have this first joint axis, and then you have the second one, and they are intersecting here. If you take the wrist, you have three intersecting axes. So when we have intersecting axes, the question is what is the common normal. So you have these axes intersecting here. What is the common normal? So we take the plane formed by I minus one and I, and take a perpendicular to that plane, and that will be a vector perpendicular to both axes. Which direction? So we have this angle, but how do we define it? Because I can take a vector in the plane or out of the plane, and that changes the direction of the angle. So we have sort of a free variable to decide in which direction we’re going to select alpha. Typically what we do is you have the base, and you’re moving toward the end-effector, so you are putting this A – the vector A, you are pointing the A towards the end-effector, so it is very intuitive to create those vectors. And once you have A defined, then you will be able to say well A is in that direction, and now I take the angle on the right hand side, or if it is in this direction, you will take it in the other direction. Okay? That’s for alpha. Now what we’re going to do next is to connect those links. So we defined the link through these two axes, the distance between them, the common normal, and the twist. But if we move further, we’re going to have another link. Now that other link will have another common normal, right? And this common normal will be between axis I and axis I plus one. So that common normal will intersect with axis I, right? It will intersect at some location with axis I. So we know this point where we have this intersection. Now what we need to do is to introduce this to other parameters that define those connections, and obviously this is perpendicular to the axis I, so I don’t know if you see this vector. How can you define this vector with respect to this line A? I used this color. Can you see it? You see this? This vector? Do you see it? And you see this vector? What are the variables that we introduce to define it?

Student:I assume the angle between the two.

Instructor (Oussama Khatib):Yeah. There is an angle between the two. That’s correct. And this angle can be found if we slide this vector through the plane. We will find it. This is going to be – so when the link, that following link is rotating, we will see this axis rotating with it and that angle increases and decreases.

And there is one more parameter, which is the distance, this offset.

Student:Describe the joint?

Student:Somewhere along the X-axis?

Instructor (Oussama Khatib):Oh, no, no. I said this is the first origin of this frame. Now there is another frame toward the next joint. Where the origin is going to be?

Student:[Inaudible] about that where [inaudible].