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.
