Introduction to robotics in topology, and defining the physical and configuration spaces

Robots are a machine which is programmed by an engineer using a computer to perform a task/s. When we tell a robot to perform certain task/s, the rules that we create for the robots to perform that tasks are called algorithm.

Now, let's look at the terms that we will use in understanding the basics of Topological Robotics.

Physical Space($\Gamma$):

Physical Space is the space where the robot/s will be performing their tasks. It is denoted as $\Gamma$. For example, as shown in Figure 6, if a robot is moving around the circle, then the physical space $\Gamma$ of that robot will be a circle $S^1$. 

Configuration Space:

• Configuration space is a space where you can abstractly define the position of robots and remove all the points where the robots cannot move which will help us to create an algorithm. 
• Mathematically, if we have $n$ robots moving on a Physical space $\Gamma$, then the configuration space will be:

$$C^n(\Gamma)= \Gamma \times \Gamma \times \Gamma \times \cdots \times \Gamma - \Delta$$ where $$\Delta= \{(x_1, x_2, x_3, \cdots,x_n) \mid \exists \, i \neq j \, where\, x_i=x_j \} $$

• The configuration space $C^1(\Gamma)$ of a robot moving on any physical space $\Gamma$ will always be $\Gamma$ because there is no cartesian product and diagonal as there is only one robot. 

Now, we will go through four examples of these two spaces.

1. A robot moving on an interval $I$.
2. Two robots moving on an interval $I$.
3. A robot moving on a circle $S^1$.
4. Two robots moving on a circle $S^1$.

A robot moving on a physical space $I$

• The configuration space $C^1(I)$ will only be $I$ as there is only one robot moving in that space. The Figure 1 showing the physical and configuration space(both are same):

Figure 1: Physical and Configuration space of a robot moving on $I$

Two robots moving a physical space $I$

• The configuration space $C^2(I)$ will be:

$$C^2(I)= I\times I- \Delta$$ where $$\Delta = \{ (x,y) \in I\times I\mid where\,  x=y\} $$

• Physical space is shown in Figure 2 and configuration space is shown in Figure 3.
• Here, the configuration space is divided into two different parts because of the diagonal $\Delta$.
• Therefore, we cannot interchange the position of the robots.

For example:

• Figure 4 shows that initial and final position of two robots moving on the $I$. 
• If we show that two positions in the configuration space, we can see from Figure 5 that it is impossible. The reason behind it is that the configuration space is divided into two spaces. 

Figure 2: Physical Space of two robots moving on $I$

Figure 3: Configuration space of two robots moving on $I$

Figure 4: Initial and Final Position of two robots moving on $I$ 

Figure 5: Initial and final position shown in the configuration space

A robot moving in a physical space $S^1$

• The configuration space $C^1(S^1)$ will be an $S^1$ as there is only one robot moving on the space. Figure 6 represents the physical and the configuration space:

Figure 6: Physical and Configuration space of a robot moving on a $S^1$

Two robots moving in a physical space $S^1$

• The configuration space $C^2(S^1)$ for this problem is: $$C^2(S^1)= S^1 \times S^1 - \Delta$$ where $$\Delta = \{ (x,y) \in S^1\times S^1\mid where \, x=y \} $$.
• We know that the cartesian product $S^1 \times S^1$ will be a torus as shown in Figure 8. But, for the configuration space, we will also have to remove the position of the robot where both overlap. 
• The physical and the configuration space is shown in Figure 7 and 9 respectively.
• Here, the configuration space is a two folded Mobius strip.
• Figure 10 shows the points where the $\Delta$ lies. 

Figure 7: Physical Space of two robots moving on a $S^1$

Figure 8: $S^1 \times S^1$ gives us torus

Figure 9: Configuration space of two robots moving on $S^1$ after cutting the $\Delta$

Figure 10: Line showing the $\Delta$

Posted by Chintan Patel.