Unified Robotics I: Actuation

Linkages

Fourbar Linkage

We will now examine a very common mechanism called a fourbar linkage.

As you can see from the figure, there are two grounded pivot points and three links labeled 2, 3, and 4 that connect them together. The fourth link, link 1, consists of the ground itself. Each of the links has a particular length. The lengths of the links, as we shall see shortly, are very important to the motion of the fourbar mechanism.

Figure 1 - Open Configuration

The connection between link 2 and link 3 - called point A, and the connection between links 3 and 4 - called point B, have particular positions in the coordinate system. These positions are going to be functions of the link lengths as well as the angles that the links form with respect to the coordinate system axes.

Figure 2 - Crossed Configuration

The particular arrangement of the links, where all of them are on or above the X axis, is called the 'Open' configuration. There is another possible configuration however and that's shown in the next figure.

This is called the Crossed configuration - because one of the links (link 3 in this case) crosses the X axis. Notice that while the angle for link two (θ2) is the same in both configurations, the other angles - those for links 3 and 4 - are different.

What is the angle for θ2

  • Show Answer
    • 0 (zero) as link 1 lies directly along the X axis in this case.

Let's focus now for a second on the open configuration. We can see that angles θ2 and θ4 are both measured with respect to the X axis. We can also see the θ3 is also measured with respect to the X axis (or at least a line parallel to it). The origin of this coordinate system is located at O2. This is also called the Global Coordinate System or GCS.

Figure 3 - GCS and LNCS -Open

We've created a secondary coordinate system with its origin at point A. This is denoted as LNCS or Local Non-rotating Coordinate System. Another coordinate system is present as well. The X axis of this coordinate system (X') lies along the AB link. It's called the Local Rotating Coordinate System or LRCS as it is attached, if you will, to the AB link and moves with it. It should be apparent that the LNCS only translates as link 2 rotates while the LRCS will translate and rotate (both with respect to the GCS).

We could apply the same analysis for the crossed configuration as well of course. Try sketching what you think that would look like and then compare it with the answer here.

  • Show Answer
    • Figure 4 - GCS and LNCS - Crossed
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