Consider transforming a robot model. If is expressed by naming specific points in , as in a boundary representation of a polygon, then each point is simply transformed from to . In this case, it is straightforward to transform the entire model using . However, there is a slight complication if the robot model is expressed using primitives, such as

This differs slightly from (3.2) because the robot is defined in (which is not necessarily ), and also is used to denote a point . Under a transformation , the primitive is transformed as

To transform the primitive completely, however, it is better to directly name points in , as opposed to . Using the fact that , this becomes

in which the inverse of appears in the right side because the original point needs to be recovered to evaluate . Therefore, it is important to be careful because either or may be required to transform the model. This will be observed in more specific contexts in some coming examples.

Steven M LaValle 2020-08-14