2.3.2. Configuration Space Representation

This video introduces representations of manifolds using minimum-coordinate explicit parametrizations and implicit representations, where the manifold is viewed as a surface embedded in a higher-dimensional space.

To represent a C-space using real numbers, we have to make some arbitrary choices. For example, to represent points on a plane, we choose a point in the space as the origin, and two orthogonal coordinate axes. With that choice, we can represent any point as a list of two coordinates, x-y. Of course our representation of the space does not change the underlying space itself. Therefore, the topology of the space is independent of our representation of the space.

If the space is "flat," like a line, a plane, or more generally an n-dimensional Euclidean space, we typically choose an origin and coordinate axes and then use coordinates to represent a point. This is what we are most familiar with. A velocity is then just the time derivative of those coordinates.

If the space is curved, however, like a sphere, we have two ways we could represent it: we could either use an explicit parametrization, which uses a minimum number of coordinates to represent the space, such as latitude and longitude for a sphere. Or we could use an IMPLICIT REPRESENTATION, which uses more coordinates, subject to constraints. An implicit representation views the n-dimensional space as embedded in a higher-dimensional Euclidean space. In the sphere example, we view the two-dimensional surface as embedded in a three-dimensional Euclidean space, and we use three Euclidean coordinates, x-y-z , subject to a single constant radius constraint. As we learned before, one constraint on three coordinates implies two degrees of freedom, that is, a two-dimensional C-space.

So how do we choose between explicit and implicit representations? An advantage of the explicit parametrization is the simplicity of a minimum number of coordinates. A disadvantage is that, because the topology of the space is different from a Euclidean space, the representation will have poor behavior at some points of the space. For example, if you walk at a constant speed along a constant latitude near the equator, your longitude changes slowly. If you do it near the North Pole, however, your longitude changes very quickly, with no upper bound as you get closer to the North Pole. The North Pole is called a SINGULARITY of the representation. Also, the moment you step over the North Pole, your longitude changes by 180 degrees. The rapidly changing coordinates and discontinuities at certain points in the space are not great properties of a representation. Keep in mind that this has nothing to do with the topology of the sphere: the sphere looks the same everywhere, at the North Pole or on the equator. It is only an issue with our representation of the sphere.

With the implicit representation using x-y-z coordinates subject to one constraint, there are no problems anywhere with discontinuities or rapidly changing coordinates. The disadvantage is the somewhat greater complexity of the representation.

Throughout this book we use implicit representations, particularly for the curved, non-Euclidean space of orientations of a rigid body. The singularity-free implicit representation we use is called the ROTATION MATRIX.

In summary, we typically do not represent configurations using a minimum set of coordinates, and we typically do not represent velocities as the time rate of change of coordinates.

In the next video, we'll learn about two types of constraints on the motion of a robot: configuration constraints and velocity constraints.