Defining frames

It was assumed so far that ${\cal A}$ is defined in ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$, but before it is transformed, it is not considered to be a subset of ${\cal W}$. The transformation $h$ places the robot in ${\cal W}$. In the coming material, it will be convenient to indicate this distinction using coordinate frames. The origin and coordinate basis vectors of ${\cal W}$ will be referred to as the world frame.^3.3Thus, any point $w \in {\cal W}$ is expressed in terms of the world frame.

The coordinates used to define ${\cal A}$ are initially expressed in the body frame, which represents the origin and coordinate basis vectors of ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$. In the case of ${\cal A}\subset {\mathbb{R}}^2$, it can be imagined that the body frame is painted on the robot. Transforming the robot is equivalent to converting its model from the body frame to the world frame. This has the effect of placing^3.4 ${\cal A}$ into ${\cal W}$ at some position and orientation. When multiple bodies are covered in Section 3.3, each body will have its own body frame, and transformations require expressing all bodies with respect to the world frame.