kinematics

kinematics - Introduction to Homogeneous Transformations...

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Page 1 of 25 Introduction to Homogeneous Transformations & Robot Kinematics Jennifer Kay, Rowan University Computer Science Department January 2005 1. Drawing 3 Dimensional Frames in 2 Dimensions We will be working in 3-D coordinates, and will label the axes x , y , and z . Figure 1 contains a sample 3-D coor- dinate frame. Because we are representing 3-D coordinate frames with 2-D drawings, we have to agree on what these drawings mean. Clearly the y axis in Figure 1 points to the right, and the z axis points up, but we have to come up with a convention for what direction the x axis is pointing. Since the three axes must be perpendicular to each other, we know that the x axis either points into the paper, or out of the paper. Most people instantly assume one or the other is the case. To be able to view both cases, it helps to look at the axes overlaid on a cube. Consider the two views of the same cube in Figure 2. In view (a) we are looking at the cube from below, in view (b) we are looking at the cube from above. Let’s try and overlay the 3-D coordinate frame from Figure 1 onto these two views. Before you turn the page, make sure you can see both views of the cube in Figure 2! x y z Figure 1. A 3-D coordinate frame. Figure 2. Two views of the same cube. The cube is missing the front side, has a solid back and sides, and patterned top and bottom. In view a we are looking at the cube from below, in view (b) we are looking at the cube from above. (a) (b) Copyright (C) 2003, 2005, Jennifer Kay. Permission is granted to make individual copies of this for educational purposes as long as this copyright notice remains intact. Permission to make multiple copies of this for educational purposes will generally be granted when an email request is sent to [email protected] (so I know who is using it). In such cases, I will require that it is distributed for free (or for the actual cost of printing / duplication) and this copy- right notice remains intact.

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Page 2 of 25 Figure 3 and Figure 4 show the same two views of the cube, this time with the 3-D coordinate frame from Figure 1 overlaid onto the cube. Note that in Figure 3 the x axis points into the paper, away from you, and in Figure 4 the x axis is pointing out of the paper towards you! For the purposes of this document, we will assume that Figure 4 shows the interpretation we will use. In other words, if you see 3 axes drawn as they are in Figure 1, you should assume that the x axis points out of the paper towards you. If you actually wanted the x axis to be pointing into the paper, you should use the illustration shown in Figure 5. Figure 3. A cube viewed from below. The edge labelled x points into the page away from you . x y z Figure 4. The same cube viewed from above. The edge labelled x points out of the page towards you .
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kinematics - Introduction to Homogeneous Transformations...

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