#### Homogeneous transformation matrices for 2D chains

We are now prepared to determine the location of each link. The location in of a point in is determined by applying the 2D homogeneous transformation matrix (3.35), (3.51)

As shown in Figure 3.10, let be the distance between the joints in . The orientation difference between and is denoted by the angle . Let represent a homogeneous transformation matrix (3.35), specialized for link for , (3.52)

This generates the following sequence of transformations:
1. Rotate counterclockwise by .
2. Translate by along the -axis.
The transformation expresses the difference between the body frame of and the body frame of . The application of moves from its body frame to the body frame of . The application of moves both and to the body frame of . By following this procedure, the location in of any point is determined by multiplying the transformation matrices to obtain (3.53)

Example 3..3 (A 2D Chain of Three Links)   To gain an intuitive understanding of these transformations, consider determining the configuration for link , as shown in Figure 3.11. Figure 3.11a shows a three-link chain in which is at its initial configuration and the other links are each offset by from the previous link. Figure 3.11b shows the frame in which the model for is initially defined. The application of causes a rotation of and a translation by . As shown in Figure 3.11c, this places in its appropriate configuration. Note that can be placed in its initial configuration, and it will be attached correctly to . The application of to the previous result places both and in their proper configurations, and can be placed in its initial configuration.  For revolute joints, the parameters are constants, and the parameters are variables. The transformed th link is represented as . In some cases, the first link might have a fixed location in the world. In this case, the revolute joints account for all degrees of freedom, yielding . For prismatic joints, the parameters are variables, instead of the parameters. It is straightforward to include both types of joints in the same kinematic chain.

Steven M LaValle 2012-04-20