4.4.2 Kinematic Chains in ${\mathbb{R}}^2$

To illustrate the concepts it will be helpful to study a simple case in detail. Let ${\cal W}= {\mathbb{R}}^2$, and suppose there is a chain of links, ${\cal A}_1$, $\ldots$, ${\cal A}_n$, as considered in Example 3.3 for $n=3$. Suppose that the first link is attached at the origin of ${\cal W}$ by a revolute joint, and every other link, ${\cal A}_i$ is attached to ${\cal A}_{i-1}$ by a revolute joint. This yields the C-space

$${\cal C}= {\mathbb{S}}^1 \times {\mathbb{S}}^1 \times \cdots \times {\mathbb{S}}^1 = {\mathbb{T}}^n,$$ (4.57)

which is the $n$-dimensional torus.