Configuration Space for Two Robots on a Circle

The Scenario

Physical space $\Gamma =  S^1$ with the two robots $x$ and $y$

    Given the scenario of two robots on a physical space $\Gamma = S^1$ where the robots cannot overlap, what is the configuration space $X$?

Solution

    One can approximately define the configuration space $X$ as a Cartesian product of the physical space $\Gamma$ and itself by the number of robots. For example, if there were three robots in a physical space $\Gamma$, then the configuration space $X$ would approximately be $\Gamma \times \Gamma \times \Gamma$. The Cartesian product only approximates the configuration space since a diagonal $\Delta$ exists in the configuration space where the robots $x$ and $y$ overlap. The configuration Space then would be defined by the Cartesian product of the physical space minus the diagonal. When the physical space $\Gamma = S^1$, the configuration space $X$ is defined by:

$X = C^2 (\Gamma) = S^1 \times S^1 - \Delta$

    The diagonal $\Delta$ is just a set of points where $x$ and $y$ overlap and can be defined as:

$\Delta = \{(x,y) \in S^1 \times S^1 | x=y\}$

    What exactly does $X$ look like, though? $S^1 \times S^1$ is a torus, but since points that lie on the diagonal cannot exist in the configuration space, $X$ is a torus with a cut made along the diagonal. What shape does this make?

A torus or $S^1 \times S^1$

    One could also express the torus as a two-dimensional sheet where the opposite edges connect. Representing $X$ is as follows:

This drawing expresses $X$. The rectangle is a sheet where the edges that have a matching number of arrows connect to form the shape of a torus; the dashed line is the diagonal $\Delta$ where there is a cut in the torus; any point $(x,y)$, in the space that does not lie on $\Delta$, correlates to a valid configuration of the two robots in the physical space $\Gamma$ 

    Since the edges of the rectangle with a matching number of arrows denotes a connection and the dashed line denotes a cut along $\Delta$, then you could cut the rectangle along the dashed line and connect a pair of matching edges, resulting in the following:

    This then shows that the configuration space $X$ is a cylinder or $S^1 \times I$ where $I = [0,1]$.

- Luis Chirinos Discua