2.3.1. Configuration Space Topology

This video introduces basic concepts in topology as applied to configuration spaces.

In addition to the number of degrees of freedom, another important property of a configuration space is its shape, or topology. Consider a plane and the surface of a sphere, for example. Both of these spaces have two dimensions, but their shape is quite different: the sphere wraps around in a way that the plane does not. This difference in shape impacts the way we use coordinates to represent the space.

We say two spaces have the same "shape," or more formally that they are topologically equivalent, if one can be smoothly deformed into the other, without cutting or gluing. A classic example is shown in this video, where the surface of a doughnut, also called a torus, is smoothly deformed into the surface of a coffee mug. These are both two-dimensional spaces. They cannot be deformed into a plane, however: that would require cutting. So the mug and the torus are topologically equivalent, but they are not equivalent to a plane. The topology of a space is a fundamental property, and it is not affected by our choice of how to represent the space with coordinates.

Some topologically distinct one-dimensional spaces are the circle, the line, and a closed interval of the line. Topologically distinct two-dimensional spaces include the plane, the surface of a sphere, the surface of a torus, and the surface of a cylinder.

Let's look at some examples of physical systems with two-dimensional C-spaces. The first is a point moving in a plane. The topology of the C-space is just a two-dimensional Euclidean space, and a configuration can be represented by two real numbers. A spherical pendulum pivots about the center of the sphere, and the topology of the C-space is the two-dimensional surface of a sphere. A configuration can be represented by latitude and longitude. The C-space of a 2R robot is a torus, and a configuration can be represented by two coordinates ranging from zero to 2 pi. And finally, the C-space of a rotating sliding knob is a cylinder, and a configuration can be represented by one real number, representing the sliding distance, and one angle between zero and 2 pi. The topology of each C-space, as you see in the middle column, does not depend on how we decide to represent the space using coordinates, whereas the representation in coordinates depends on an arbitrary choice, such as where we define the zero angle for each joint of the 2R robot.

Let's focus on the 2R robot. The topology of the C-space is a torus. We can represent the torus using the two joint angle coordinates, ranging between 0 and 2 pi. The space of coordinates is obtained from the torus by cutting the torus once to get a cylinder, then again to get a square subset of the plane. Because of this cutting, which means that the square and the torus do not have the same topology, even if the configuration on the torus moves smoothly, the coordinate representation changes discontinuously at 0 and 2 pi. In this video, you can see that as the robot moves, the coordinate representation jumps suddenly from one edge of the coordinate square to the other.

Now let's focus on the rotating and sliding knob. Its C-space is a cylinder, due to one linear joint and one rotational joint. We can cut this cylinder once to get our coordinate representation, a flat subset of the two-dimensional plane. The angle coordinate is discontinuous at 0 and 2 pi. As the robot moves in this video, you see the discontinuity in the representation of the knob angle.

Finally, let's look at the spherical pendulum. It has a spherical C-space, and we can see its representation as a subset of the plane. Each of the points on the top line segment of the representation correspond to the same point, the North Pole of the sphere, and each of the points on the bottom line segment correspond to the South Pole. This video shows the changing representation as the spherical pendulum moves.

In summary, C-spaces of the same dimension can have different topologies. In the next video, we discuss different ways to represent C-spaces that are not flat Euclidean spaces.