Common joints for ${\cal W}= {\mathbb{R}}^2$

First consider the simplest case in which there is a 2D tree of links for which every link has only two points at which revolute joints may be attached. This corresponds to Figure 3.21a. A single link is designated as the root, ${\cal A}_1$, of the tree. To determine the transformation of a body, ${\cal A}_i$, in the tree, the tools from Section 3.3.1 are directly applied to the chain of bodies that connects ${\cal A}_i$ to ${\cal A}_1$ while ignoring all other bodies. Each link contributes a $\theta _i$ to the total degrees of freedom of the tree. This case seems quite straightforward; unfortunately, it is not this easy in general.