How to draw the following configuration space (manifold)?

Question

I am studying configuration spaces for robots. For example, the configuration of a two-linked robot can be described as $\mathbf{q}=(\theta_1,\theta_2)$ and the configuration space is a torus ($\mathbf{Q} = \mathrm{S}^1 \times \mathrm{S}^1$). If I have a omnidirectional 2D robot, my configuration is $\mathbf{q} = (x,y)$ with $\mathrm{Q} = \mathrm{R}^2$, i.e., a plane. In a one-wheeled robot, my configuration is $\mathbf{q} = (x,y,\theta)$, i.e., I need the position and the orientation to describe the state of the robot. Thus, the configuration space is $\mathrm{Q} = \mathrm{R}^2 \times \mathrm{S}^1$. Is it possible to draw this manifold? How can I geometrically interpret this space? Thank you!

Answer 1

I have a representation to propose. Is it interesting for your purpose, I don't know...

Consider the set of 3D oriented straight lines that are parallel to vertical plane xOz and intersect base plane xOy.

They can be parameterized by $(x,y,\theta)$, with $(x,y)$ their point of intersection with xOy and angle $\theta$ with respect to horizontal plane.

Edit: One can give of this manifold slightly different representations by replacing the oriented lines

• by their unit vector, or,

• (probably better) by line segments connecting a point $(x_1,y,0)$ to a point $(x_2,y,1)$, with the same $y$, the angle being obtained as $\theta=\operatorname{atan}(1/(x_2-x_1)$. This representation has the advantage to make the connection with affine representation mentionned here and there. A drawback, angles $\pi/2$ and $3\pi/2$ aren't represented; they have to be "added manually" (!)

Answer 2

You can build $\Bbb{R}^2 \times S^1$ by first building $\Bbb{R}^2 \times [0, 1]$ and then identifying the top and bottom faces via $(x, y, 0) \sim (x, y, 1)$. (In this space, if you float up to through the top face, you exit out of the bottom face directly vertically below where you started.)