Introduction to Functions, Continuity of functions and Section Function

To understand the Continuity of Sections, let's first look at the meaning of functions and its continuity.

Functions:

Functions are something which takes an input from the set of points known as domain and gives exactly one output from that point. The set of all the points in the output is called range. In other words, each point in the domain map exactly to one unique point. Here, the output is dependent on what input we give to the functions. Let's see an example below.

If the function $f(x)$ is equal to the square of $x$, then
$$ f(x) = x^2 $$
Here, if we input different value of $x$ into the function $f(x)$, then we should get one unique value for that input. If $x=3$, then $f(x)=9$, and when $x=(-4)$, then $f(x)=16$. We can also draw the representation of funtions in graph. For the above example, the graph of the functions is as shown below.

Figure 1: Example of a function $f(x) = x^2$

Continuity of Functions:

If we look back at the definition of a continuous function in Calculus, then these three arguments should be true to a function to be continuous at point $a$:

1. The function value at $a$ exists, i.e. $f(a)$ exists. 
2. $\lim_{x\to a} f(x) $ exists.
3. $\lim_{x\to a}f(x) = f(a) $

We can also look at this concept from different perspective. 

Let's say that we have a function $f(x)$. For that function to be continuous from any range $(f(a)- \varepsilon, f(a)+ \varepsilon)$, there exists an interval $(a- \delta, a+\delta)$ such that for all $x \in (a- \delta, a+\delta) $, there exists $f(x) \in (f(a)- \varepsilon, f(a) + \varepsilon)$. If this scenerio is true, then the function $f(x)$ is continuous at $a$. If this is true for all the points in $f(x)$, then the whole function is continuous. 

For example, Figure 2 shows that the function is continuous while Figure 3 shows a discontinuous function at point $a$. 

Figure 2: A continuous function $f(x)$

Figure 3: A discontinuous function $f(x)$

Continuity of the Section Function:

If we recall from one of our last posts, then the definition of Section function is as follows:

$$s: X \times X \to PX $$ 

We also said that this is one of the most important elements in the field of Topological Robotics as it gives us the path between two points. 

Here, we will use the same method that we used to find the continuity of function to find the continuity of section. 

Let's take a neighborhood of a point $(x,y)$ on the Cartesian product and assume its radius as $\delta$. Let's take another point $(x',y')$ in the neighborhood of $(x,y)$. Using Section function, we get a path between $x$ and $y$, and another path between $x'$ and $y'$. If these two paths are closer to each other, then the section is continuous. Figure 4 is the example of a continuous section, while Figure 5 is an example of a discontinuous section. 

Figure 4: An example for continuous section

Figure 5: An example for discontinuous section

Posted by Chintan Patel.

Path, Path space, evaluation and section function

Definition of Path function:

Mathematically, a path is a function of an interval $I$ to space $X$ where $I$ is from $[0,1]$. A path can also be a connection between some points in space $X$. To understand it more clearly, let's take an example of Global Positioning System(GPS). In GPS, when a person is finding a navigation between two points on a map, the result that comes out is the path between that two points. In topology, a path is a function defined by $\alpha : I \to X$ where the initial point is $\alpha(0)=a$ and the final point is $\alpha(1)=b$ as shown in Figure 1.

Figure 1(a): Simple example of a path between a and b

Figure 1(b): Path in $GPS$

 Figure 1: Examples of different path

Definition of Path Space($PX$):

Path Space is a space which contains all the path in space $X$. It is written as:

$$PX=\{ \alpha \mid \alpha \, \mbox{is a path in} \, X \} $$

Definition of Evaluation function:

The evaluation function is the function which takes a path from the path space $PX$ as an input and gives a pair of initial and final position in the Cartesian Product $X \times X$ as an output. In other words, it is a function whose domain is the path space $PX$ and the codomain is the Cartesian Product $X \times X$. It is written as:

$$ev: PX \to X \times X $$

If we take a path as an input, then the output is the initial and the final position is shown in the Cartesian Product. For example, if $\alpha$ is the path taken as an input, then the output is the initial point $\alpha (0) $ and final point $\alpha (1)$ of that path.  

Definition of Section Functions:

The section is a one-sided inverse of the evaluation function because it takes two points on the Cartesian product as an input and gives a path between that two points as an output. It is written as:

$$s: X \times X \to PX \, \mbox{such that} \, ev \circ s= identity $$

This function is the main function in defining our motion planning problem as it takes two points and gives us the path between them, and that is what we need to move a robot from one position to another. To solve the motion planning problem means to find the section. The section can be continuous or discontinuous or may not exist. 

Identity Function:

First, you take two points as an input as of section and that gives you the path $\alpha$ between them. Second, we take that path $\alpha$ as an input as of evaluation function and it gives us the initial and final points. Therefore, these two functions give us the identity function. 

Posted by Chintan Patel. 

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.