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 ,

This generates the following sequence of transformations:

- Rotate counterclockwise by .
- Translate by along the -axis.

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