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

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

which is the $ n$-dimensional torus.


Steven M LaValle 2012-04-20